Calculus Examples

$\int_{- 2}^{2} p {(\sqrt{4 - x^{2}})}^{2} d x$

Step 1

Rewrite ${(\sqrt{4 - x^{2}})}^{2}$ as $4 - x^{2}$ .

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Step 1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{4 - x^{2}}$ as ${(4 - x^{2})}^{\frac{1}{2}}$ .

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

Step 1.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\int_{- 2}^{2} p {(4 - x^{2})}^{\frac{1}{2} \cdot 2} d x$

Step 1.3

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

$\int_{- 2}^{2} p {(4 - x^{2})}^{\frac{2}{2}} d x$

Step 1.4

Cancel the common factor of $2$ .

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Step 1.4.1

Cancel the common factor.

$\int_{- 2}^{2} p {(4 - x^{2})}^{\frac{2}{2}} d x$

Step 1.4.2

Rewrite the expression.

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

Step 1.5

Simplify.

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

Step 2

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

$p \int_{- 2}^{2} 4 - x^{2} d x$

Step 3

Split the single integral into multiple integrals.

$p (\int_{- 2}^{2} 4 d x + \int_{- 2}^{2} - x^{2} d x)$

Step 4

Apply the constant rule.

$p (4 x]_{- 2}^{2} + \int_{- 2}^{2} - x^{2} d x)$

Step 5

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

$p (4 x]_{- 2}^{2} - \int_{- 2}^{2} x^{2} d x)$

Step 6

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

$p (4 x]_{- 2}^{2} - (\frac{1}{3} x^{3}]_{- 2}^{2}))$

Step 7

Simplify the answer.

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Step 7.1

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

$p (4 x]_{- 2}^{2} - (\frac{x^{3}}{3}]_{- 2}^{2}))$

Step 7.2

Substitute and simplify.

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Step 7.2.1

Evaluate $4 x$ at $2$ and at $- 2$ .

$p ((4 \cdot 2) - 4 \cdot - 2 - (\frac{x^{3}}{3}]_{- 2}^{2}))$

Step 7.2.2

Evaluate $\frac{x^{3}}{3}$ at $2$ and at $- 2$ .

$p (4 \cdot 2 - 4 \cdot - 2 - (\frac{2^{3}}{3} - \frac{{(- 2)}^{3}}{3}))$

Step 7.2.3

Simplify.

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Step 7.2.3.1

Multiply $4$ by $2$ .

$p (8 - 4 \cdot - 2 - (\frac{2^{3}}{3} - \frac{{(- 2)}^{3}}{3}))$

Step 7.2.3.2

Multiply $- 4$ by $- 2$ .

$p (8 + 8 - (\frac{2^{3}}{3} - \frac{{(- 2)}^{3}}{3}))$

Step 7.2.3.3

Add $8$ and $8$ .

$p (16 - (\frac{2^{3}}{3} - \frac{{(- 2)}^{3}}{3}))$

Step 7.2.3.4

Raise $2$ to the power of $3$ .

$p (16 - (\frac{8}{3} - \frac{{(- 2)}^{3}}{3}))$

Step 7.2.3.5

Raise $- 2$ to the power of $3$ .

$p (16 - (\frac{8}{3} - \frac{- 8}{3}))$

Step 7.2.3.6

Move the negative in front of the fraction.

$p (16 - (\frac{8}{3} - - \frac{8}{3}))$

Step 7.2.3.7

Multiply $- 1$ by $- 1$ .

$p (16 - (\frac{8}{3} + 1 (\frac{8}{3})))$

Step 7.2.3.8

Multiply $\frac{8}{3}$ by $1$ .

$p (16 - (\frac{8}{3} + \frac{8}{3}))$

Step 7.2.3.9

Combine the numerators over the common denominator.

$p (16 - \frac{8 + 8}{3})$

Step 7.2.3.10

Add $8$ and $8$ .

$p (16 - \frac{16}{3})$

Step 7.2.3.11

To write $16$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$p (16 \cdot \frac{3}{3} - \frac{16}{3})$

Step 7.2.3.12

Combine $16$ and $\frac{3}{3}$ .

$p (\frac{16 \cdot 3}{3} - \frac{16}{3})$

Step 7.2.3.13

Combine the numerators over the common denominator.

$p \frac{16 \cdot 3 - 16}{3}$

Step 7.2.3.14

Simplify the numerator.

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Step 7.2.3.14.1

Multiply $16$ by $3$ .

$p \frac{48 - 16}{3}$

Step 7.2.3.14.2

Subtract $16$ from $48$ .

$p \frac{32}{3}$

Step 7.2.3.15

Combine $p$ and $\frac{32}{3}$ .

$\frac{p \cdot 32}{3}$

Step 7.2.3.16

Move $32$ to the left of $p$ .

$\frac{32 p}{3}$

Step 7.3

Reorder terms.

$\frac{32}{3} p$

Step 8

Combine $\frac{32}{3}$ and $p$ .

$\frac{32 p}{3}$

Step 9