Calculus Examples

$x - 2 y = - 5$ , $x^{2} + y^{2} = 25$

Step 1

Solve by substitution to find the intersection between the curves.

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Step 1.1

Add $2 y$ to both sides of the equation.

$x = - 5 + 2 y$

$x^{2} + y^{2} = 25$

Step 1.2

Replace all occurrences of $x$ with $- 5 + 2 y$ in each equation.

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Step 1.2.1

Replace all occurrences of $x$ in $x^{2} + y^{2} = 25$ with $- 5 + 2 y$ .

${(- 5 + 2 y)}^{2} + y^{2} = 25$

$x = - 5 + 2 y$

Step 1.2.2

Simplify the left side.

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Step 1.2.2.1

Simplify ${(- 5 + 2 y)}^{2} + y^{2}$ .

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Step 1.2.2.1.1

Simplify each term.

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Step 1.2.2.1.1.1

Rewrite ${(- 5 + 2 y)}^{2}$ as $(- 5 + 2 y) (- 5 + 2 y)$ .

$(- 5 + 2 y) (- 5 + 2 y) + y^{2} = 25$

$x = - 5 + 2 y$

Step 1.2.2.1.1.2

Expand $(- 5 + 2 y) (- 5 + 2 y)$ using the FOIL Method.

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Step 1.2.2.1.1.2.1

Apply the distributive property.

$- 5 (- 5 + 2 y) + 2 y (- 5 + 2 y) + y^{2} = 25$

$x = - 5 + 2 y$

Step 1.2.2.1.1.2.2

Apply the distributive property.

$- 5 \cdot - 5 - 5 (2 y) + 2 y (- 5 + 2 y) + y^{2} = 25$

$x = - 5 + 2 y$

Step 1.2.2.1.1.2.3

Apply the distributive property.

$- 5 \cdot - 5 - 5 (2 y) + 2 y \cdot - 5 + 2 y (2 y) + y^{2} = 25$

$x = - 5 + 2 y$

$- 5 \cdot - 5 - 5 (2 y) + 2 y \cdot - 5 + 2 y (2 y) + y^{2} = 25$

$x = - 5 + 2 y$

Step 1.2.2.1.1.3

Simplify and combine like terms.

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Step 1.2.2.1.1.3.1

Simplify each term.

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Step 1.2.2.1.1.3.1.1

Multiply $- 5$ by $- 5$ .

$25 - 5 (2 y) + 2 y \cdot - 5 + 2 y (2 y) + y^{2} = 25$

$x = - 5 + 2 y$

Step 1.2.2.1.1.3.1.2

Multiply $2$ by $- 5$ .

$25 - 10 y + 2 y \cdot - 5 + 2 y (2 y) + y^{2} = 25$

$x = - 5 + 2 y$

Step 1.2.2.1.1.3.1.3

Multiply $- 5$ by $2$ .

$25 - 10 y - 10 y + 2 y (2 y) + y^{2} = 25$

$x = - 5 + 2 y$

Step 1.2.2.1.1.3.1.4

Rewrite using the commutative property of multiplication.

$25 - 10 y - 10 y + 2 \cdot (2 y \cdot y) + y^{2} = 25$

$x = - 5 + 2 y$

Step 1.2.2.1.1.3.1.5

Multiply $y$ by $y$ by adding the exponents.

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Step 1.2.2.1.1.3.1.5.1

Move $y$ .

$25 - 10 y - 10 y + 2 \cdot (2 (y \cdot y)) + y^{2} = 25$

$x = - 5 + 2 y$

Step 1.2.2.1.1.3.1.5.2

Multiply $y$ by $y$ .

$25 - 10 y - 10 y + 2 \cdot (2 y^{2}) + y^{2} = 25$

$x = - 5 + 2 y$

$25 - 10 y - 10 y + 2 \cdot (2 y^{2}) + y^{2} = 25$

$x = - 5 + 2 y$

Step 1.2.2.1.1.3.1.6

Multiply $2$ by $2$ .

$25 - 10 y - 10 y + 4 y^{2} + y^{2} = 25$

$x = - 5 + 2 y$

$25 - 10 y - 10 y + 4 y^{2} + y^{2} = 25$

$x = - 5 + 2 y$

Step 1.2.2.1.1.3.2

Subtract $10 y$ from $- 10 y$ .

$25 - 20 y + 4 y^{2} + y^{2} = 25$

$x = - 5 + 2 y$

$25 - 20 y + 4 y^{2} + y^{2} = 25$

$x = - 5 + 2 y$

$25 - 20 y + 4 y^{2} + y^{2} = 25$

$x = - 5 + 2 y$

Step 1.2.2.1.2

Add $4 y^{2}$ and $y^{2}$ .

$25 - 20 y + 5 y^{2} = 25$

$x = - 5 + 2 y$

$25 - 20 y + 5 y^{2} = 25$

$x = - 5 + 2 y$

$25 - 20 y + 5 y^{2} = 25$

$x = - 5 + 2 y$

$25 - 20 y + 5 y^{2} = 25$

$x = - 5 + 2 y$

Step 1.3

Solve for $y$ in $25 - 20 y + 5 y^{2} = 25$ .

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Step 1.3.1

Subtract $25$ from both sides of the equation.

$25 - 20 y + 5 y^{2} - 25 = 0$

$x = - 5 + 2 y$

Step 1.3.2

Combine the opposite terms in $25 - 20 y + 5 y^{2} - 25$ .

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Step 1.3.2.1

Subtract $25$ from $25$ .

$- 20 y + 5 y^{2} + 0 = 0$

$x = - 5 + 2 y$

Step 1.3.2.2

Add $- 20 y + 5 y^{2}$ and $0$ .

$- 20 y + 5 y^{2} = 0$

$x = - 5 + 2 y$

$- 20 y + 5 y^{2} = 0$

$x = - 5 + 2 y$

Step 1.3.3

Factor $- 5 y$ out of $- 20 y + 5 y^{2}$ .

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Step 1.3.3.1

Reorder $- 20 y$ and $5 y^{2}$ .

$5 y^{2} - 20 y = 0$

$x = - 5 + 2 y$

Step 1.3.3.2

Factor $- 5 y$ out of $5 y^{2}$ .

$- 5 y (- y) - 20 y = 0$

$x = - 5 + 2 y$

Step 1.3.3.3

Factor $- 5 y$ out of $- 20 y$ .

$- 5 y (- y) - 5 y \cdot 4 = 0$

$x = - 5 + 2 y$

Step 1.3.3.4

Factor $- 5 y$ out of $- 5 y (- y) - 5 y (4)$ .

$- 5 y (- y + 4) = 0$

$x = - 5 + 2 y$

$- 5 y (- y + 4) = 0$

$x = - 5 + 2 y$

Step 1.3.4

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$y = 0$

$- y + 4 = 0$

$x = - 5 + 2 y$

Step 1.3.5

Set $y$ equal to $0$ .

$y = 0$

$x = - 5 + 2 y$

Step 1.3.6

Set $- y + 4$ equal to $0$ and solve for $y$ .

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Step 1.3.6.1

Set $- y + 4$ equal to $0$ .

$- y + 4 = 0$

$x = - 5 + 2 y$

Step 1.3.6.2

Solve $- y + 4 = 0$ for $y$ .

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Step 1.3.6.2.1

Subtract $4$ from both sides of the equation.

$- y = - 4$

$x = - 5 + 2 y$

Step 1.3.6.2.2

Divide each term in $- y = - 4$ by $- 1$ and simplify.

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Step 1.3.6.2.2.1

Divide each term in $- y = - 4$ by $- 1$ .

$\frac{- y}{- 1} = \frac{- 4}{- 1}$

$x = - 5 + 2 y$

Step 1.3.6.2.2.2

Simplify the left side.

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Step 1.3.6.2.2.2.1

Dividing two negative values results in a positive value.

$\frac{y}{1} = \frac{- 4}{- 1}$

$x = - 5 + 2 y$

Step 1.3.6.2.2.2.2

Divide $y$ by $1$ .

$y = \frac{- 4}{- 1}$

$x = - 5 + 2 y$

$y = \frac{- 4}{- 1}$

$x = - 5 + 2 y$

Step 1.3.6.2.2.3

Simplify the right side.

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Step 1.3.6.2.2.3.1

Divide $- 4$ by $- 1$ .

$y = 4$

$x = - 5 + 2 y$

$y = 4$

$x = - 5 + 2 y$

$y = 4$

$x = - 5 + 2 y$

$y = 4$

$x = - 5 + 2 y$

$y = 4$

$x = - 5 + 2 y$

Step 1.3.7

The final solution is all the values that make $- 5 y (- y + 4) = 0$ true.

$y = 0, 4$

$x = - 5 + 2 y$

$y = 0, 4$

$x = - 5 + 2 y$

Step 1.4

Replace all occurrences of $y$ with $0$ in each equation.

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Step 1.4.1

Replace all occurrences of $y$ in $x = - 5 + 2 y$ with $0$ .

$x = - 5 + 2 (0)$

$y = 0$

Step 1.4.2

Simplify the right side.

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Step 1.4.2.1

Simplify $- 5 + 2 (0)$ .

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Step 1.4.2.1.1

Multiply $2$ by $0$ .

$x = - 5 + 0$

$y = 0$

Step 1.4.2.1.2

Add $- 5$ and $0$ .

$x = - 5$

$y = 0$

$x = - 5$

$y = 0$

$x = - 5$

$y = 0$

$x = - 5$

$y = 0$

Step 1.5

Replace all occurrences of $y$ with $4$ in each equation.

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Step 1.5.1

Replace all occurrences of $y$ in $x = - 5 + 2 y$ with $4$ .

$x = - 5 + 2 (4)$

$y = 4$

Step 1.5.2

Simplify the right side.

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Step 1.5.2.1

Simplify $- 5 + 2 (4)$ .

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Step 1.5.2.1.1

Multiply $2$ by $4$ .

$x = - 5 + 8$

$y = 4$

Step 1.5.2.1.2

Add $- 5$ and $8$ .

$x = 3$

$y = 4$

$x = 3$

$y = 4$

$x = 3$

$y = 4$

$x = 3$

$y = 4$

Step 1.6

The solution to the system is the complete set of ordered pairs that are valid solutions.

$(- 5, 0)$

$(3, 4)$

$(- 5, 0)$

$(3, 4)$

Step 2

Add $2 y$ to both sides of the equation.

$x = - 5 + 2 y$

Step 3

Solve $x^{2} + y^{2} = 25$ in terms of $x$ .

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Step 3.1

Subtract $y^{2}$ from both sides of the equation.

$x^{2} = 25 - y^{2}$

Step 3.2

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$x = \pm \sqrt{25 - y^{2}}$

Step 3.3

Simplify $\pm \sqrt{25 - y^{2}}$ .

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Step 3.3.1

Rewrite $25$ as $5^{2}$ .

$x = \pm \sqrt{5^{2} - y^{2}}$

Step 3.3.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 5$ and $b = y$ .

$x = \pm \sqrt{(5 + y) (5 - y)}$

Step 3.4

The complete solution is the result of both the positive and negative portions of the solution.

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Step 3.4.1

First, use the positive value of the $\pm$ to find the first solution.

$x = \sqrt{(5 + y) (5 - y)}$

Step 3.4.2

Next, use the negative value of the $\pm$ to find the second solution.

$x = - \sqrt{(5 + y) (5 - y)}$

Step 3.4.3

The complete solution is the result of both the positive and negative portions of the solution.

$x = \sqrt{(5 + y) (5 - y)}$

$x = - \sqrt{(5 + y) (5 - y)}$

$x = \sqrt{(5 + y) (5 - y)}$

$x = - \sqrt{(5 + y) (5 - y)}$

$x = \sqrt{(5 + y) (5 - y)}$

$x = - \sqrt{(5 + y) (5 - y)}$

Step 4

The area of the region between the curves is defined as the integral of the upper curve minus the integral of the lower curve over each region. The regions are determined by the intersection points of the curves. This can be done algebraically or graphically.

$A r e a = \int_{0}^{4} \sqrt{(5 + y) (5 - y)} d y - \int_{0}^{4} - 5 + 2 y d y$

Step 5

Integrate to find the area between $0$ and $4$ .

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Step 5.1

Combine the integrals into a single integral.

$\int_{0}^{4} \sqrt{(5 + y) (5 - y)} - (- 5 + 2 y) d y$

Step 5.2

Simplify each term.

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Step 5.2.1

Apply the distributive property.

$\sqrt{(5 + y) (5 - y)} - - 5 - (2 y)$

Step 5.2.2

Multiply $- 1$ by $- 5$ .

$\sqrt{(5 + y) (5 - y)} + 5 - (2 y)$

Step 5.2.3

Multiply $2$ by $- 1$ .

$\sqrt{(5 + y) (5 - y)} + 5 - 2 y$

$\int_{0}^{4} \sqrt{(5 + y) (5 - y)} + 5 - 2 y d y$

Step 5.3

Split the single integral into multiple integrals.

$\int_{0}^{4} \sqrt{(5 + y) (5 - y)} d y + \int_{0}^{4} 5 d y + \int_{0}^{4} - 2 y d y$

Step 5.4

Complete the square.

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Step 5.4.1

Simplify the expression.

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Step 5.4.1.1

Expand $(5 + y) (5 - y)$ using the FOIL Method.

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Step 5.4.1.1.1

Apply the distributive property.

$5 (5 - y) + y (5 - y)$

Step 5.4.1.1.2

Apply the distributive property.

$5 \cdot 5 + 5 (- y) + y (5 - y)$

Step 5.4.1.1.3

Apply the distributive property.

$5 \cdot 5 + 5 (- y) + y \cdot 5 + y (- y)$

Step 5.4.1.2

Simplify and combine like terms.

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Step 5.4.1.2.1

Simplify each term.

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Step 5.4.1.2.1.1

Multiply $5$ by $5$ .

$25 + 5 (- y) + y \cdot 5 + y (- y)$

Step 5.4.1.2.1.2

Multiply $- 1$ by $5$ .

$25 - 5 y + y \cdot 5 + y (- y)$

Step 5.4.1.2.1.3

Move $5$ to the left of $y$ .

$25 - 5 y + 5 \cdot y + y (- y)$

Step 5.4.1.2.1.4

Rewrite using the commutative property of multiplication.

$25 - 5 y + 5 y - y \cdot y$

Step 5.4.1.2.1.5

Multiply $y$ by $y$ by adding the exponents.

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Step 5.4.1.2.1.5.1

Move $y$ .

$25 - 5 y + 5 y - (y \cdot y)$

Step 5.4.1.2.1.5.2

Multiply $y$ by $y$ .

$25 - 5 y + 5 y - y^{2}$

Step 5.4.1.2.2

Add $- 5 y$ and $5 y$ .

$25 + 0 - y^{2}$

Step 5.4.1.2.3

Add $25$ and $0$ .

$25 - y^{2}$

Step 5.4.1.3

Reorder $25$ and $- y^{2}$ .

$- y^{2} + 25$

Step 5.4.2

Use the form $a x^{2} + b x + c$ , to find the values of $a$ , $b$ , and $c$ .

$a = - 1$

$b = 0$

$c = 25$

Step 5.4.3

Consider the vertex form of a parabola.

$a {(x + d)}^{2} + e$

Step 5.4.4

Find the value of $d$ using the formula $d = \frac{b}{2 a}$ .

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Step 5.4.4.1

Substitute the values of $a$ and $b$ into the formula $d = \frac{b}{2 a}$ .

$d = \frac{0}{2 \cdot - 1}$

Step 5.4.4.2

Simplify the right side.

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Step 5.4.4.2.1

Cancel the common factor of $0$ and $2$ .

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Step 5.4.4.2.1.1

Factor $2$ out of $0$ .

$d = \frac{2 (0)}{2 \cdot - 1}$

Step 5.4.4.2.1.2

Move the negative one from the denominator of $\frac{0}{- 1}$ .

$d = - 1 \cdot 0$

Step 5.4.4.2.2

Rewrite $- 1 \cdot 0$ as $- 0$ .

$d = - 0$

Step 5.4.4.2.3

Multiply $- 1$ by $0$ .

$d = 0$

Step 5.4.5

Find the value of $e$ using the formula $e = c - \frac{b^{2}}{4 a}$ .

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Step 5.4.5.1

Substitute the values of $c$ , $b$ and $a$ into the formula $e = c - \frac{b^{2}}{4 a}$ .

$e = 25 - \frac{0^{2}}{4 \cdot - 1}$

Step 5.4.5.2

Simplify the right side.

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Step 5.4.5.2.1

Simplify each term.

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Step 5.4.5.2.1.1

Raising $0$ to any positive power yields $0$ .

$e = 25 - \frac{0}{4 \cdot - 1}$

Step 5.4.5.2.1.2

Multiply $4$ by $- 1$ .

$e = 25 - \frac{0}{- 4}$

Step 5.4.5.2.1.3

Divide $0$ by $- 4$ .

$e = 25 - 0$

Step 5.4.5.2.1.4

Multiply $- 1$ by $0$ .

$e = 25 + 0$

Step 5.4.5.2.2

Add $25$ and $0$ .

$e = 25$

Step 5.4.6

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $- {(y + 0)}^{2} + 25$ .

$\int_{0}^{4} \sqrt{- {(y + 0)}^{2} + 25} d y + \int_{0}^{4} 5 d y + \int_{0}^{4} - 2 y d y$

Step 5.5

Let $u_{1} = y + 0$ . Then $d u_{1} = d y$ . Rewrite using $u_{1}$ and $d$ $u_{1}$ .

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Step 5.5.1

Let $u_{1} = y + 0$ . Find $\frac{d u_{1}}{d y}$ .

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Step 5.5.1.1

Differentiate $y + 0$ .

$\frac{d}{d y} [y + 0]$

Step 5.5.1.2

By the Sum Rule, the derivative of $y + 0$ with respect to $y$ is $\frac{d}{d y} [y] + \frac{d}{d y} [0]$ .

$\frac{d}{d y} [y] + \frac{d}{d y} [0]$

Step 5.5.1.3

Differentiate using the Power Rule which states that $\frac{d}{d y} [y^{n}]$ is $n y^{n - 1}$ where $n = 1$ .

$1 + \frac{d}{d y} [0]$

Step 5.5.1.4

Since $0$ is constant with respect to $y$ , the derivative of $0$ with respect to $y$ is $0$ .

$1 + 0$

Step 5.5.1.5

Add $1$ and $0$ .

$1$

Step 5.5.2

Substitute the lower limit in for $y$ in $u_{1} = y + 0$ .

$u_{lower} = 0 + 0$

Step 5.5.3

Add $0$ and $0$ .

$u_{lower} = 0$

Step 5.5.4

Substitute the upper limit in for $y$ in $u_{1} = y + 0$ .

$u_{upper} = 4 + 0$

Step 5.5.5

Add $4$ and $0$ .

$u_{upper} = 4$

Step 5.5.6

The values found for $u_{lower}$ and $u_{upper}$ will be used to evaluate the definite integral.

$u_{lower} = 0$

$u_{upper} = 4$

Step 5.5.7

Rewrite the problem using $u_{1}$ , $d u_{1}$ , and the new limits of integration.

$\int_{0}^{4} \sqrt{- {u_{1}}^{2} + 25} d u_{1} + \int_{0}^{4} 5 d y + \int_{0}^{4} - 2 y d y$

Step 5.6

Let $u_{1} = 5 \sin (t)$ , where $- \frac{π}{2} \leq t \leq \frac{π}{2}$ . Then $d u_{1} = 5 \cos (t) d t$ . Note that since $- \frac{π}{2} \leq t \leq \frac{π}{2}$ , $5 \cos (t)$ is positive.

$\int_{0}^{0.92729521} \sqrt{- {(5 \sin (t))}^{2} + 25} (5 \cos (t)) d t + \int_{0}^{4} 5 d y + \int_{0}^{4} - 2 y d y$

Step 5.7

Simplify terms.

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Step 5.7.1

Simplify $\sqrt{- {(5 \sin (t))}^{2} + 25}$ .

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Step 5.7.1.1

Simplify each term.

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Step 5.7.1.1.1

Apply the product rule to $5 \sin (t)$ .

$\int_{0}^{0.92729521} \sqrt{- (5^{2} \sin^{2} (t)) + 25} (5 \cos (t)) d t + \int_{0}^{4} 5 d y + \int_{0}^{4} - 2 y d y$

Step 5.7.1.1.2

Raise $5$ to the power of $2$ .

$\int_{0}^{0.92729521} \sqrt{- (25 \sin^{2} (t)) + 25} (5 \cos (t)) d t + \int_{0}^{4} 5 d y + \int_{0}^{4} - 2 y d y$

Step 5.7.1.1.3

Multiply $25$ by $- 1$ .

$\int_{0}^{0.92729521} \sqrt{- 25 \sin^{2} (t) + 25} (5 \cos (t)) d t + \int_{0}^{4} 5 d y + \int_{0}^{4} - 2 y d y$

Step 5.7.1.2

Reorder $- 25 \sin^{2} (t)$ and $25$ .

$\int_{0}^{0.92729521} \sqrt{25 - 25 \sin^{2} (t)} (5 \cos (t)) d t + \int_{0}^{4} 5 d y + \int_{0}^{4} - 2 y d y$

Step 5.7.1.3

Factor $25$ out of $25$ .

$\int_{0}^{0.92729521} \sqrt{25 (1) - 25 \sin^{2} (t)} (5 \cos (t)) d t + \int_{0}^{4} 5 d y + \int_{0}^{4} - 2 y d y$

Step 5.7.1.4

Factor $25$ out of $- 25 \sin^{2} (t)$ .

$\int_{0}^{0.92729521} \sqrt{25 (1) + 25 (- \sin^{2} (t))} (5 \cos (t)) d t + \int_{0}^{4} 5 d y + \int_{0}^{4} - 2 y d y$

Step 5.7.1.5

Factor $25$ out of $25 (1) + 25 (- \sin^{2} (t))$ .

$\int_{0}^{0.92729521} \sqrt{25 (1 - \sin^{2} (t))} (5 \cos (t)) d t + \int_{0}^{4} 5 d y + \int_{0}^{4} - 2 y d y$

Step 5.7.1.6

Apply pythagorean identity.

$\int_{0}^{0.92729521} \sqrt{25 \cos^{2} (t)} (5 \cos (t)) d t + \int_{0}^{4} 5 d y + \int_{0}^{4} - 2 y d y$

Step 5.7.1.7

Rewrite $25 \cos^{2} (t)$ as ${(5 \cos (t))}^{2}$ .

$\int_{0}^{0.92729521} \sqrt{{(5 \cos (t))}^{2}} (5 \cos (t)) d t + \int_{0}^{4} 5 d y + \int_{0}^{4} - 2 y d y$

Step 5.7.1.8

Pull terms out from under the radical, assuming positive real numbers.

$\int_{0}^{0.92729521} 5 \cos (t) (5 \cos (t)) d t + \int_{0}^{4} 5 d y + \int_{0}^{4} - 2 y d y$

Step 5.7.2

Simplify.

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Step 5.7.2.1

Multiply $5$ by $5$ .

$\int_{0}^{0.92729521} 25 \cos (t) \cos (t) d t + \int_{0}^{4} 5 d y + \int_{0}^{4} - 2 y d y$

Step 5.7.2.2

Raise $\cos (t)$ to the power of $1$ .

$\int_{0}^{0.92729521} 25 (\cos^{1} (t) \cos (t)) d t + \int_{0}^{4} 5 d y + \int_{0}^{4} - 2 y d y$

Step 5.7.2.3

Raise $\cos (t)$ to the power of $1$ .

$\int_{0}^{0.92729521} 25 (\cos^{1} (t) \cos^{1} (t)) d t + \int_{0}^{4} 5 d y + \int_{0}^{4} - 2 y d y$

Step 5.7.2.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\int_{0}^{0.92729521} 25 {\cos (t)}^{1 + 1} d t + \int_{0}^{4} 5 d y + \int_{0}^{4} - 2 y d y$

Step 5.7.2.5

Add $1$ and $1$ .

$\int_{0}^{0.92729521} 25 \cos^{2} (t) d t + \int_{0}^{4} 5 d y + \int_{0}^{4} - 2 y d y$

Step 5.8

Since $25$ is constant with respect to $t$ , move $25$ out of the integral.

$25 \int_{0}^{0.92729521} \cos^{2} (t) d t + \int_{0}^{4} 5 d y + \int_{0}^{4} - 2 y d y$

Step 5.9

Use the half-angle formula to rewrite $\cos^{2} (t)$ as $\frac{1 + \cos (2 t)}{2}$ .

$25 \int_{0}^{0.92729521} \frac{1 + \cos (2 t)}{2} d t + \int_{0}^{4} 5 d y + \int_{0}^{4} - 2 y d y$

Step 5.10

Since $\frac{1}{2}$ is constant with respect to $t$ , move $\frac{1}{2}$ out of the integral.

$25 (\frac{1}{2} \int_{0}^{0.92729521} 1 + \cos (2 t) d t) + \int_{0}^{4} 5 d y + \int_{0}^{4} - 2 y d y$

Step 5.11

Combine $\frac{1}{2}$ and $25$ .

$\frac{25}{2} \int_{0}^{0.92729521} 1 + \cos (2 t) d t + \int_{0}^{4} 5 d y + \int_{0}^{4} - 2 y d y$

Step 5.12

Split the single integral into multiple integrals.

$\frac{25}{2} (\int_{0}^{0.92729521} d t + \int_{0}^{0.92729521} \cos (2 t) d t) + \int_{0}^{4} 5 d y + \int_{0}^{4} - 2 y d y$

Step 5.13

Apply the constant rule.

$\frac{25}{2} (t]_{0}^{0.92729521} + \int_{0}^{0.92729521} \cos (2 t) d t) + \int_{0}^{4} 5 d y + \int_{0}^{4} - 2 y d y$

Step 5.14

Let $u_{2} = 2 t$ . Then $d u_{2} = 2 d t$ , so $\frac{1}{2} d u_{2} = d t$ . Rewrite using $u_{2}$ and $d$ $u_{2}$ .

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Step 5.14.1

Let $u_{2} = 2 t$ . Find $\frac{d u_{2}}{d t}$ .

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Step 5.14.1.1

Differentiate $2 t$ .

$\frac{d}{d t} [2 t]$

Step 5.14.1.2

Since $2$ is constant with respect to $t$ , the derivative of $2 t$ with respect to $t$ is $2 \frac{d}{d t} [t]$ .

$2 \frac{d}{d t} [t]$

Step 5.14.1.3

Differentiate using the Power Rule which states that $\frac{d}{d t} [t^{n}]$ is $n t^{n - 1}$ where $n = 1$ .

$2 \cdot 1$

Step 5.14.1.4

Multiply $2$ by $1$ .

$2$

Step 5.14.2

Substitute the lower limit in for $t$ in $u_{2} = 2 t$ .

$u_{lower} = 2 \cdot 0$

Step 5.14.3

Multiply $2$ by $0$ .

$u_{lower} = 0$

Step 5.14.4

Substitute the upper limit in for $t$ in $u_{2} = 2 t$ .

$u_{upper} = 2 \cdot 0.92729521$

Step 5.14.5

Multiply $2$ by $0.92729521$ .

$u_{upper} = 1.85459043$

Step 5.14.6

The values found for $u_{lower}$ and $u_{upper}$ will be used to evaluate the definite integral.

$u_{lower} = 0$

$u_{upper} = 1.85459043$

Step 5.14.7

Rewrite the problem using $u_{2}$ , $d u_{2}$ , and the new limits of integration.

$\frac{25}{2} (t]_{0}^{0.92729521} + \int_{0}^{1.85459043} \cos (u_{2}) \frac{1}{2} d u_{2}) + \int_{0}^{4} 5 d y + \int_{0}^{4} - 2 y d y$

Step 5.15

Combine $\cos (u_{2})$ and $\frac{1}{2}$ .

$\frac{25}{2} (t]_{0}^{0.92729521} + \int_{0}^{1.85459043} \frac{\cos (u_{2})}{2} d u_{2}) + \int_{0}^{4} 5 d y + \int_{0}^{4} - 2 y d y$

Step 5.16

Since $\frac{1}{2}$ is constant with respect to $u_{2}$ , move $\frac{1}{2}$ out of the integral.

$\frac{25}{2} (t]_{0}^{0.92729521} + \frac{1}{2} \int_{0}^{1.85459043} \cos (u_{2}) d u_{2}) + \int_{0}^{4} 5 d y + \int_{0}^{4} - 2 y d y$

Step 5.17

The integral of $\cos (u_{2})$ with respect to $u_{2}$ is $\sin (u_{2})$ .

$\frac{25}{2} (t]_{0}^{0.92729521} + \frac{1}{2} \sin (u_{2})]_{0}^{1.85459043}) + \int_{0}^{4} 5 d y + \int_{0}^{4} - 2 y d y$

Step 5.18

Apply the constant rule.

$\frac{25}{2} (t]_{0}^{0.92729521} + \frac{1}{2} \sin (u_{2})]_{0}^{1.85459043}) + 5 y]_{0}^{4} + \int_{0}^{4} - 2 y d y$

Step 5.19

Since $- 2$ is constant with respect to $y$ , move $- 2$ out of the integral.

$\frac{25}{2} (t]_{0}^{0.92729521} + \frac{1}{2} \sin (u_{2})]_{0}^{1.85459043}) + 5 y]_{0}^{4} - 2 \int_{0}^{4} y d y$

Step 5.20

By the Power Rule, the integral of $y$ with respect to $y$ is $\frac{1}{2} y^{2}$ .

$\frac{25}{2} (t]_{0}^{0.92729521} + \frac{1}{2} \sin (u_{2})]_{0}^{1.85459043}) + 5 y]_{0}^{4} - 2 (\frac{1}{2} y^{2}]_{0}^{4})$

Step 5.21

Combine $\frac{1}{2}$ and $y^{2}$ .

$\frac{25}{2} (t]_{0}^{0.92729521} + \frac{1}{2} \sin (u_{2})]_{0}^{1.85459043}) + 5 y]_{0}^{4} - 2 (\frac{y^{2}}{2}]_{0}^{4})$

Step 5.22

Substitute and simplify.

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Step 5.22.1

Evaluate $t$ at $0.92729521$ and at $0$ .

$\frac{25}{2} ((0.92729521) + 0 + \frac{1}{2} \sin (u_{2})]_{0}^{1.85459043}) + 5 y]_{0}^{4} - 2 (\frac{y^{2}}{2}]_{0}^{4})$

Step 5.22.2

Evaluate $\sin (u_{2})$ at $1.85459043$ and at $0$ .

$\frac{25}{2} ((0.92729521) + 0 + \frac{1}{2} ((\sin (1.85459043)) - \sin (0))) + 5 y]_{0}^{4} - 2 (\frac{y^{2}}{2}]_{0}^{4})$

Step 5.22.3

Evaluate $5 y$ at $4$ and at $0$ .

$\frac{25}{2} ((0.92729521) + 0 + \frac{1}{2} ((\sin (1.85459043)) - \sin (0))) + (5 \cdot 4) - 5 \cdot 0 - 2 (\frac{y^{2}}{2}]_{0}^{4})$

Step 5.22.4

Evaluate $\frac{y^{2}}{2}$ at $4$ and at $0$ .

$\frac{25}{2} ((0.92729521) + 0 + \frac{1}{2} ((\sin (1.85459043)) - \sin (0))) + (5 \cdot 4) - 5 \cdot 0 - 2 ((\frac{4^{2}}{2}) - \frac{0^{2}}{2})$

Step 5.22.5

Simplify.

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Step 5.22.5.1

Add $0.92729521$ and $0$ .

$\frac{25}{2} (0.92729521 + \frac{1}{2} ((\sin (1.85459043)) - \sin (0))) + (5 \cdot 4) - 5 \cdot 0 - 2 ((\frac{4^{2}}{2}) - \frac{0^{2}}{2})$

Step 5.22.5.2

Multiply $5$ by $4$ .

$\frac{25}{2} (0.92729521 + \frac{1}{2} (\sin (1.85459043) - \sin (0))) + 20 - 5 \cdot 0 - 2 ((\frac{4^{2}}{2}) - \frac{0^{2}}{2})$

Step 5.22.5.3

Multiply $- 5$ by $0$ .

$\frac{25}{2} (0.92729521 + \frac{1}{2} (\sin (1.85459043) - \sin (0))) + 20 + 0 - 2 ((\frac{4^{2}}{2}) - \frac{0^{2}}{2})$

Step 5.22.5.4

Add $20$ and $0$ .

$\frac{25}{2} (0.92729521 + \frac{1}{2} (\sin (1.85459043) - \sin (0))) + 20 - 2 ((\frac{4^{2}}{2}) - \frac{0^{2}}{2})$

Step 5.22.5.5

Raise $4$ to the power of $2$ .

$\frac{25}{2} (0.92729521 + \frac{1}{2} (\sin (1.85459043) - \sin (0))) + 20 - 2 (\frac{16}{2} - \frac{0^{2}}{2})$

Step 5.22.5.6

Cancel the common factor of $16$ and $2$ .

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Step 5.22.5.6.1

Factor $2$ out of $16$ .

$\frac{25}{2} (0.92729521 + \frac{1}{2} (\sin (1.85459043) - \sin (0))) + 20 - 2 (\frac{2 \cdot 8}{2} - \frac{0^{2}}{2})$

Step 5.22.5.6.2

Cancel the common factors.

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Step 5.22.5.6.2.1

Factor $2$ out of $2$ .

$\frac{25}{2} (0.92729521 + \frac{1}{2} (\sin (1.85459043) - \sin (0))) + 20 - 2 (\frac{2 \cdot 8}{2 (1)} - \frac{0^{2}}{2})$

Step 5.22.5.6.2.2

Cancel the common factor.

$\frac{25}{2} (0.92729521 + \frac{1}{2} (\sin (1.85459043) - \sin (0))) + 20 - 2 (\frac{2 \cdot 8}{2 \cdot 1} - \frac{0^{2}}{2})$

Step 5.22.5.6.2.3

Rewrite the expression.

$\frac{25}{2} (0.92729521 + \frac{1}{2} (\sin (1.85459043) - \sin (0))) + 20 - 2 (\frac{8}{1} - \frac{0^{2}}{2})$

Step 5.22.5.6.2.4

Divide $8$ by $1$ .

$\frac{25}{2} (0.92729521 + \frac{1}{2} (\sin (1.85459043) - \sin (0))) + 20 - 2 (8 - \frac{0^{2}}{2})$

Step 5.22.5.7

Raising $0$ to any positive power yields $0$ .

$\frac{25}{2} (0.92729521 + \frac{1}{2} (\sin (1.85459043) - \sin (0))) + 20 - 2 (8 - \frac{0}{2})$

Step 5.22.5.8

Cancel the common factor of $0$ and $2$ .

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Step 5.22.5.8.1

Factor $2$ out of $0$ .

$\frac{25}{2} (0.92729521 + \frac{1}{2} (\sin (1.85459043) - \sin (0))) + 20 - 2 (8 - \frac{2 (0)}{2})$

Step 5.22.5.8.2

Cancel the common factors.

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Step 5.22.5.8.2.1

Factor $2$ out of $2$ .

$\frac{25}{2} (0.92729521 + \frac{1}{2} (\sin (1.85459043) - \sin (0))) + 20 - 2 (8 - \frac{2 \cdot 0}{2 \cdot 1})$

Step 5.22.5.8.2.2

Cancel the common factor.

$\frac{25}{2} (0.92729521 + \frac{1}{2} (\sin (1.85459043) - \sin (0))) + 20 - 2 (8 - \frac{2 \cdot 0}{2 \cdot 1})$

Step 5.22.5.8.2.3

Rewrite the expression.

$\frac{25}{2} (0.92729521 + \frac{1}{2} (\sin (1.85459043) - \sin (0))) + 20 - 2 (8 - \frac{0}{1})$

Step 5.22.5.8.2.4

Divide $0$ by $1$ .

$\frac{25}{2} (0.92729521 + \frac{1}{2} (\sin (1.85459043) - \sin (0))) + 20 - 2 (8 - 0)$

Step 5.22.5.9

Multiply $- 1$ by $0$ .

$\frac{25}{2} (0.92729521 + \frac{1}{2} (\sin (1.85459043) - \sin (0))) + 20 - 2 (8 + 0)$

Step 5.22.5.10

Add $8$ and $0$ .

$\frac{25}{2} (0.92729521 + \frac{1}{2} (\sin (1.85459043) - \sin (0))) + 20 - 2 \cdot 8$

Step 5.22.5.11

Multiply $- 2$ by $8$ .

$\frac{25}{2} (0.92729521 + \frac{1}{2} (\sin (1.85459043) - \sin (0))) + 20 - 16$

Step 5.22.5.12

Subtract $16$ from $20$ .

$\frac{25}{2} (0.92729521 + \frac{1}{2} (\sin (1.85459043) - \sin (0))) + 4$

Step 5.23

Simplify.

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Step 5.23.1

The exact value of $\sin (0)$ is $0$ .

$\frac{25}{2} (0.92729521 + \frac{1}{2} (\sin (1.85459043) - 0)) + 4$

Step 5.23.2

Multiply $- 1$ by $0$ .

$\frac{25}{2} (0.92729521 + \frac{1}{2} (\sin (1.85459043) + 0)) + 4$

Step 5.23.3

Add $\sin (1.85459043)$ and $0$ .

$\frac{25}{2} (0.92729521 + \frac{1}{2} \sin (1.85459043)) + 4$

Step 5.23.4

Combine $\frac{1}{2}$ and $\sin (1.85459043)$ .

$\frac{25}{2} (0.92729521 + \frac{\sin (1.85459043)}{2}) + 4$

Step 5.23.5

Add $0.92729521$ and $\frac{\sin (1.85459043)}{2}$ .

$\frac{25}{2} \cdot 1.40729521 + 4$

Step 5.23.6

Combine $\frac{25}{2}$ and $1.40729521$ .

$\frac{25 \cdot 1.40729521}{2} + 4$

Step 5.23.7

Multiply $25$ by $1.40729521$ .

$\frac{35.18238045}{2} + 4$

Step 5.23.8

To write $4$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{35.18238045}{2} + 4 \cdot \frac{2}{2}$

Step 5.23.9

Combine $4$ and $\frac{2}{2}$ .

$\frac{35.18238045}{2} + \frac{4 \cdot 2}{2}$

Step 5.23.10

Combine the numerators over the common denominator.

$\frac{35.18238045 + 4 \cdot 2}{2}$

Step 5.23.11

Multiply $4$ by $2$ .

$\frac{35.18238045 + 8}{2}$

Step 5.23.12

Add $35.18238045$ and $8$ .

$\frac{43.18238045}{2}$

Step 5.24

Divide $43.18238045$ by $2$ .

$21.59119022$

Step 6