Calculus Examples

$y = 4 x^{2}$ , $x = 1$ , $x = 4$

Step 1

Solve by substitution to find the intersection between the curves.

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Step 1.1

Eliminate the equal sides of each equation and combine.

$4 x^{2} = 0$

Step 1.2

Solve $4 x^{2} = 0$ for $x$ .

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Step 1.2.1

Divide each term in $4 x^{2} = 0$ by $4$ and simplify.

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Step 1.2.1.1

Divide each term in $4 x^{2} = 0$ by $4$ .

$\frac{4 x^{2}}{4} = \frac{0}{4}$

Step 1.2.1.2

Simplify the left side.

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Step 1.2.1.2.1

Cancel the common factor of $4$ .

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Step 1.2.1.2.1.1

Cancel the common factor.

$\frac{4 x^{2}}{4} = \frac{0}{4}$

Step 1.2.1.2.1.2

Divide $x^{2}$ by $1$ .

$x^{2} = \frac{0}{4}$

Step 1.2.1.3

Simplify the right side.

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Step 1.2.1.3.1

Divide $0$ by $4$ .

$x^{2} = 0$

Step 1.2.2

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

$x = \pm \sqrt{0}$

Step 1.2.3

Simplify $\pm \sqrt{0}$ .

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Step 1.2.3.1

Rewrite $0$ as $0^{2}$ .

$x = \pm \sqrt{0^{2}}$

Step 1.2.3.2

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

$x = \pm 0$

Step 1.2.3.3

Plus or minus $0$ is $0$ .

$x = 0$

Step 1.3

Substitute $0$ for $x$ .

$y = 0$

Step 1.4

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

$(0, 0)$

Step 2

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.

$A r e a = \int_{1}^{4} 4 x^{2} d x - \int_{1}^{4} 0 d x$

Step 3

Integrate to find the area between $1$ and $4$ .

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Step 3.1

Combine the integrals into a single integral.

$\int_{1}^{4} 4 x^{2} - (0) d x$

Step 3.2

Subtract $0$ from $4 x^{2}$ .

$\int_{1}^{4} 4 x^{2} d x$

Step 3.3

Since $4$ is constant with respect to $x$ , move $4$ out of the integral.

$4 \int_{1}^{4} x^{2} d x$

Step 3.4

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

$4 (\frac{1}{3} x^{3}]_{1}^{4})$

Step 3.5

Simplify the answer.

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Step 3.5.1

Combine $\frac{1}{3}$ and $x^{3}$ .

$4 (\frac{x^{3}}{3}]_{1}^{4})$

Step 3.5.2

Substitute and simplify.

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Step 3.5.2.1

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

$4 ((\frac{4^{3}}{3}) - \frac{1^{3}}{3})$

Step 3.5.2.2

Simplify.

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Step 3.5.2.2.1

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

$4 (\frac{64}{3} - \frac{1^{3}}{3})$

Step 3.5.2.2.2

One to any power is one.

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

Step 3.5.2.2.3

Combine the numerators over the common denominator.

$4 \frac{64 - 1}{3}$

Step 3.5.2.2.4

Subtract $1$ from $64$ .

$4 (\frac{63}{3})$

Step 3.5.2.2.5

Cancel the common factor of $63$ and $3$ .

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Step 3.5.2.2.5.1

Factor $3$ out of $63$ .

$4 \frac{3 \cdot 21}{3}$

Step 3.5.2.2.5.2

Cancel the common factors.

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Step 3.5.2.2.5.2.1

Factor $3$ out of $3$ .

$4 \frac{3 \cdot 21}{3 (1)}$

Step 3.5.2.2.5.2.2

Cancel the common factor.

$4 \frac{3 \cdot 21}{3 \cdot 1}$

Step 3.5.2.2.5.2.3

Rewrite the expression.

$4 (\frac{21}{1})$

Step 3.5.2.2.5.2.4

Divide $21$ by $1$ .

$4 \cdot 21$

Step 3.5.2.2.6

Multiply $4$ by $21$ .

$84$

Step 4