Calculus Examples

$\int_{0}^{5} π {(\frac{20 - 4 x}{5})}^{2} d x$

Step 1

Since $π$ is constant with respect to $x$ , move $π$ out of the integral.

$π \int_{0}^{5} {(\frac{20 - 4 x}{5})}^{2} d x$

Step 2

Let $u = \frac{20 - 4 x}{5}$ . Then $d u = - \frac{4}{5} d x$ , so $- \frac{5}{4} d u = d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 2.1

Let $u = \frac{20 - 4 x}{5}$ . Find $\frac{d u}{d x}$ .

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Step 2.1.1

Differentiate $\frac{20 - 4 x}{5}$ .

$\frac{d}{d x} [\frac{20 - 4 x}{5}]$

Step 2.1.2

Factor $4$ out of $20 - 4 x$ .

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Step 2.1.2.1

Factor $4$ out of $20$ .

$\frac{d}{d x} [\frac{4 (5) - 4 x}{5}]$

Step 2.1.2.2

Factor $4$ out of $- 4 x$ .

$\frac{d}{d x} [\frac{4 (5) + 4 (- x)}{5}]$

Step 2.1.2.3

Factor $4$ out of $4 (5) + 4 (- x)$ .

$\frac{d}{d x} [\frac{4 (5 - x)}{5}]$

Step 2.1.3

Since $\frac{4}{5}$ is constant with respect to $x$ , the derivative of $\frac{4 (5 - x)}{5}$ with respect to $x$ is $\frac{4}{5} \frac{d}{d x} [5 - x]$ .

$\frac{4}{5} \frac{d}{d x} [5 - x]$

Step 2.1.4

By the Sum Rule, the derivative of $5 - x$ with respect to $x$ is $\frac{d}{d x} [5] + \frac{d}{d x} [- x]$ .

$\frac{4}{5} (\frac{d}{d x} [5] + \frac{d}{d x} [- x])$

Step 2.1.5

Since $5$ is constant with respect to $x$ , the derivative of $5$ with respect to $x$ is $0$ .

$\frac{4}{5} (0 + \frac{d}{d x} [- x])$

Step 2.1.6

Add $0$ and $\frac{d}{d x} [- x]$ .

$\frac{4}{5} \frac{d}{d x} [- x]$

Step 2.1.7

Since $- 1$ is constant with respect to $x$ , the derivative of $- x$ with respect to $x$ is $- \frac{d}{d x} [x]$ .

$\frac{4}{5} (- \frac{d}{d x} [x])$

Step 2.1.8

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

$\frac{4}{5} (- 1 \cdot 1)$

Step 2.1.9

Combine fractions.

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Step 2.1.9.1

Multiply $- 1$ by $1$ .

$\frac{4}{5} \cdot - 1$

Step 2.1.9.2

Combine $\frac{4}{5}$ and $- 1$ .

$\frac{4 \cdot - 1}{5}$

Step 2.1.9.3

Simplify the expression.

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Step 2.1.9.3.1

Multiply $4$ by $- 1$ .

$\frac{- 4}{5}$

Step 2.1.9.3.2

Move the negative in front of the fraction.

$- \frac{4}{5}$

Step 2.2

Substitute the lower limit in for $x$ in $u = \frac{20 - 4 x}{5}$ .

$u_{lower} = \frac{20 - 4 \cdot 0}{5}$

Step 2.3

Simplify.

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Step 2.3.1

Cancel the common factor of $20 - 4 \cdot 0$ and $5$ .

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Step 2.3.1.1

Rewrite $20$ as $- 1 (- 20)$ .

$u_{lower} = \frac{- 1 (- 20) - 4 \cdot 0}{5}$

Step 2.3.1.2

Factor $- 1$ out of $- 4 \cdot 0$ .

$u_{lower} = \frac{- 1 (- 20) - (4 \cdot 0)}{5}$

Step 2.3.1.3

Factor $- 1$ out of $- 1 (- 20) - (4 \cdot 0)$ .

$u_{lower} = \frac{- 1 (- 20 + 4 \cdot 0)}{5}$

Step 2.3.1.4

Reorder terms.

$u_{lower} = \frac{- 1 (- 20 + 0 \cdot 4)}{5}$

Step 2.3.1.5

Factor $5$ out of $- 1 (- 20 + 0 \cdot 4)$ .

$u_{lower} = \frac{5 (- 1 (- 4 + 0 \cdot 4))}{5}$

Step 2.3.1.6

Cancel the common factors.

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Step 2.3.1.6.1

Factor $5$ out of $5$ .

$u_{lower} = \frac{5 (- 1 (- 4 + 0 \cdot 4))}{5 (1)}$

Step 2.3.1.6.2

Cancel the common factor.

$u_{lower} = \frac{5 (- 1 (- 4 + 0 \cdot 4))}{5 \cdot 1}$

Step 2.3.1.6.3

Rewrite the expression.

$u_{lower} = \frac{- 1 (- 4 + 0 \cdot 4)}{1}$

Step 2.3.1.6.4

Divide $- 1 (- 4 + 0 \cdot 4)$ by $1$ .

$u_{lower} = - 1 (- 4 + 0 \cdot 4)$

Step 2.3.2

Rewrite $- 1 (- 4 + 0 \cdot 4)$ as $- (- 4 + 0 \cdot 4)$ .

$u_{lower} = - (- 4 + 0 \cdot 4)$

Step 2.3.3

Multiply $0$ by $4$ .

$u_{lower} = - (- 4 + 0)$

Step 2.3.4

Add $- 4$ and $0$ .

$u_{lower} = - - 4$

Step 2.3.5

Multiply $- 1$ by $- 4$ .

$u_{lower} = 4$

Step 2.4

Substitute the upper limit in for $x$ in $u = \frac{20 - 4 x}{5}$ .

$u_{upper} = \frac{20 - 4 \cdot 5}{5}$

Step 2.5

Simplify.

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Step 2.5.1

Cancel the common factor of $20 - 4 \cdot 5$ and $5$ .

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Step 2.5.1.1

Rewrite $20$ as $- 1 (- 20)$ .

$u_{upper} = \frac{- 1 (- 20) - 4 \cdot 5}{5}$

Step 2.5.1.2

Factor $- 1$ out of $- 4 \cdot 5$ .

$u_{upper} = \frac{- 1 (- 20) - (4 \cdot 5)}{5}$

Step 2.5.1.3

Factor $- 1$ out of $- 1 (- 20) - (4 \cdot 5)$ .

$u_{upper} = \frac{- 1 (- 20 + 4 \cdot 5)}{5}$

Step 2.5.1.4

Reorder terms.

$u_{upper} = \frac{- 1 (4 \cdot 5 - 20)}{5}$

Step 2.5.1.5

Factor $5$ out of $- 1 (4 \cdot 5 - 20)$ .

$u_{upper} = \frac{5 (- 1 (4 - 4))}{5}$

Step 2.5.1.6

Cancel the common factors.

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Step 2.5.1.6.1

Factor $5$ out of $5$ .

$u_{upper} = \frac{5 (- 1 (4 - 4))}{5 (1)}$

Step 2.5.1.6.2

Cancel the common factor.

$u_{upper} = \frac{5 (- 1 (4 - 4))}{5 \cdot 1}$

Step 2.5.1.6.3

Rewrite the expression.

$u_{upper} = \frac{- 1 (4 - 4)}{1}$

Step 2.5.1.6.4

Divide $- 1 (4 - 4)$ by $1$ .

$u_{upper} = - 1 (4 - 4)$

Step 2.5.2

Rewrite $- 1 (4 - 4)$ as $- (4 - 4)$ .

$u_{upper} = - (4 - 4)$

Step 2.5.3

Subtract $4$ from $4$ .

$u_{upper} = - 0$

Step 2.5.4

Multiply $- 1$ by $0$ .

$u_{upper} = 0$

Step 2.6

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

$u_{lower} = 4$

$u_{upper} = 0$

Step 2.7

Rewrite the problem using $u$ , $d u$ , and the new limits of integration.

$π \int_{4}^{0} - u^{2} \frac{- 1}{- \frac{4}{5}} d u$

Step 3

Simplify.

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Step 3.1

Dividing two negative values results in a positive value.

$π \int_{4}^{0} - u^{2} \frac{1}{\frac{4}{5}} d u$

Step 3.2

Multiply by the reciprocal of the fraction to divide by $\frac{4}{5}$ .

$π \int_{4}^{0} - u^{2} (1 (\frac{5}{4})) d u$

Step 3.3

Multiply $\frac{5}{4}$ by $1$ .

$π \int_{4}^{0} - u^{2} \frac{5}{4} d u$

Step 3.4

Combine $\frac{5}{4}$ and $u^{2}$ .

$π \int_{4}^{0} - \frac{5 u^{2}}{4} d u$

Step 4

Since $- 1$ is constant with respect to $u$ , move $- 1$ out of the integral.

$π (- \int_{4}^{0} \frac{5 u^{2}}{4} d u)$

Step 5

Since $\frac{5}{4}$ is constant with respect to $u$ , move $\frac{5}{4}$ out of the integral.

$π (- (\frac{5}{4} \int_{4}^{0} u^{2} d u))$

Step 6

Simplify.

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Step 6.1

Combine $π$ and $\frac{5}{4}$ .

$- \frac{π \cdot 5}{4} \int_{4}^{0} u^{2} d u$

Step 6.2

Move $5$ to the left of $π$ .

$- \frac{5 π}{4} \int_{4}^{0} u^{2} d u$

Step 7

By the Power Rule, the integral of $u^{2}$ with respect to $u$ is $\frac{1}{3} u^{3}$ .

$- \frac{5 π}{4} \frac{1}{3} u^{3}]_{4}^{0}$

Step 8

Substitute and simplify.

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Step 8.1

Evaluate $\frac{1}{3} u^{3}$ at $0$ and at $4$ .

$- \frac{5 π}{4} ((\frac{1}{3} \cdot 0^{3}) - \frac{1}{3} \cdot 4^{3})$

Step 8.2

Simplify.

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Step 8.2.1

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

$- \frac{5 π}{4} (\frac{1}{3} \cdot 0 - \frac{1}{3} \cdot 4^{3})$

Step 8.2.2

Multiply $\frac{1}{3}$ by $0$ .

$- \frac{5 π}{4} (0 - \frac{1}{3} \cdot 4^{3})$

Step 8.2.3

Raise $4$ to the power of $3$ .

$- \frac{5 π}{4} (0 - \frac{1}{3} \cdot 64)$

Step 8.2.4

Multiply $64$ by $- 1$ .

$- \frac{5 π}{4} (0 - 64 (\frac{1}{3}))$

Step 8.2.5

Combine $- 64$ and $\frac{1}{3}$ .

$- \frac{5 π}{4} (0 + \frac{- 64}{3})$

Step 8.2.6

Move the negative in front of the fraction.

$- \frac{5 π}{4} (0 - \frac{64}{3})$

Step 8.2.7

Subtract $\frac{64}{3}$ from $0$ .

$- \frac{5 π}{4} (- \frac{64}{3})$

Step 8.2.8

Multiply $- 1$ by $- 1$ .

$1 \frac{5 π}{4} \frac{64}{3}$

Step 8.2.9

Multiply $\frac{5 π}{4}$ by $1$ .

$\frac{5 π}{4} \cdot \frac{64}{3}$

Step 8.2.10

Multiply $\frac{5 π}{4}$ by $\frac{64}{3}$ .

$\frac{5 π \cdot 64}{4 \cdot 3}$

Step 8.2.11

Multiply $64$ by $5$ .

$\frac{320 π}{4 \cdot 3}$

Step 8.2.12

Multiply $4$ by $3$ .

$\frac{320 π}{12}$

Step 8.2.13

Cancel the common factor of $320$ and $12$ .

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Step 8.2.13.1

Factor $4$ out of $320 π$ .

$\frac{4 (80 π)}{12}$

Step 8.2.13.2

Cancel the common factors.

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Step 8.2.13.2.1

Factor $4$ out of $12$ .

$\frac{4 (80 π)}{4 (3)}$

Step 8.2.13.2.2

Cancel the common factor.

$\frac{4 (80 π)}{4 \cdot 3}$

Step 8.2.13.2.3

Rewrite the expression.

$\frac{80 π}{3}$

Step 9

The result can be shown in multiple forms.

Exact Form:

$\frac{80 π}{3}$

Decimal Form:

$83.77580409 \dots$

Step 10