Calculus Examples

$\int_{0}^{a} 8 x \sqrt{a^{2} - x^{2}} d x$

Step 1

Since $8$ is constant with respect to $x$ , move $8$ out of the integral.

$8 \int_{0}^{a} x \sqrt{a^{2} - x^{2}} d x$

Step 2

Let $u = a^{2} - x^{2}$ . Then $d u = - 2 x d x$ , so $- \frac{1}{2} d u = x d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 2.1

Let $u = a^{2} - x^{2}$ . Find $\frac{d u}{d x}$ .

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Step 2.1.1

Differentiate $a^{2} - x^{2}$ .

$\frac{d}{d x} [a^{2} - x^{2}]$

Step 2.1.2

Differentiate.

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Step 2.1.2.1

By the Sum Rule, the derivative of $a^{2} - x^{2}$ with respect to $x$ is $\frac{d}{d x} [a^{2}] + \frac{d}{d x} [- x^{2}]$ .

$\frac{d}{d x} [a^{2}] + \frac{d}{d x} [- x^{2}]$

Step 2.1.2.2

Since $a^{2}$ is constant with respect to $x$ , the derivative of $a^{2}$ with respect to $x$ is $0$ .

$0 + \frac{d}{d x} [- x^{2}]$

Step 2.1.3

Evaluate $\frac{d}{d x} [- x^{2}]$ .

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Step 2.1.3.1

Since $- 1$ is constant with respect to $x$ , the derivative of $- x^{2}$ with respect to $x$ is $- \frac{d}{d x} [x^{2}]$ .

$0 - \frac{d}{d x} [x^{2}]$

Step 2.1.3.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 2$ .

$0 - (2 x)$

Step 2.1.3.3

Multiply $2$ by $- 1$ .

$0 - 2 x$

Step 2.1.4

Subtract $2 x$ from $0$ .

$- 2 x$

Step 2.2

Substitute the lower limit in for $x$ in $u = a^{2} - x^{2}$ .

$u_{lower} = a^{2} - 0^{2}$

Step 2.3

Simplify.

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Step 2.3.1

Simplify each term.

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Step 2.3.1.1

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

$u_{lower} = a^{2} - 0$

Step 2.3.1.2

Multiply $- 1$ by $0$ .

$u_{lower} = a^{2} + 0$

Step 2.3.2

Add $a^{2}$ and $0$ .

$u_{lower} = a^{2}$

Step 2.4

Substitute the upper limit in for $x$ in $u = a^{2} - x^{2}$ .

$u_{upper} = a^{2} - a^{2}$

Step 2.5

Subtract $a^{2}$ from $a^{2}$ .

$u_{upper} = 0$

Step 2.6

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

$u_{lower} = a^{2}$

$u_{upper} = 0$

Step 2.7

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

$8 \int_{a^{2}}^{0} \sqrt{u} \frac{1}{- 2} d u$

Step 3

Simplify.

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Step 3.1

Move the negative in front of the fraction.

$8 \int_{a^{2}}^{0} \sqrt{u} (- \frac{1}{2}) d u$

Step 3.2

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

$8 \int_{a^{2}}^{0} - \frac{\sqrt{u}}{2} d u$

Step 4

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

$8 (- \int_{a^{2}}^{0} \frac{\sqrt{u}}{2} d u)$

Step 5

Multiply $- 1$ by $8$ .

$- 8 \int_{a^{2}}^{0} \frac{\sqrt{u}}{2} d u$

Step 6

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

$- 8 (\frac{1}{2} \int_{a^{2}}^{0} \sqrt{u} d u)$

Step 7

Simplify the expression.

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Step 7.1

Simplify.

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Step 7.1.1

Combine $\frac{1}{2}$ and $- 8$ .

$\frac{- 8}{2} \int_{a^{2}}^{0} \sqrt{u} d u$

Step 7.1.2

Cancel the common factor of $- 8$ and $2$ .

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Step 7.1.2.1

Factor $2$ out of $- 8$ .

$\frac{2 \cdot - 4}{2} \int_{a^{2}}^{0} \sqrt{u} d u$

Step 7.1.2.2

Cancel the common factors.

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Step 7.1.2.2.1

Factor $2$ out of $2$ .

$\frac{2 \cdot - 4}{2 (1)} \int_{a^{2}}^{0} \sqrt{u} d u$

Step 7.1.2.2.2

Cancel the common factor.

$\frac{2 \cdot - 4}{2 \cdot 1} \int_{a^{2}}^{0} \sqrt{u} d u$

Step 7.1.2.2.3

Rewrite the expression.

$\frac{- 4}{1} \int_{a^{2}}^{0} \sqrt{u} d u$

Step 7.1.2.2.4

Divide $- 4$ by $1$ .

$- 4 \int_{a^{2}}^{0} \sqrt{u} d u$

Step 7.2

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

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

Step 8

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

$- 4 (\frac{2}{3} u^{\frac{3}{2}}]_{a^{2}}^{0})$

Step 9

Combine $\frac{2}{3}$ and $u^{\frac{3}{2}}$ .

$- 4 (\frac{2 u^{\frac{3}{2}}}{3}]_{a^{2}}^{0})$

Step 10

Substitute and simplify.

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Step 10.1

Evaluate $\frac{2 u^{\frac{3}{2}}}{3}$ at $0$ and at $a^{2}$ .

$- 4 ((\frac{2 \cdot 0^{\frac{3}{2}}}{3}) - \frac{2 {(a^{2})}^{\frac{3}{2}}}{3})$

Step 10.2

Simplify.

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Step 10.2.1

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

$- 4 (\frac{2 \cdot {(0^{2})}^{\frac{3}{2}}}{3} - \frac{2 {(a^{2})}^{\frac{3}{2}}}{3})$

Step 10.2.2

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

$- 4 (\frac{2 \cdot 0^{2 (\frac{3}{2})}}{3} - \frac{2 {(a^{2})}^{\frac{3}{2}}}{3})$

Step 10.2.3

Cancel the common factor of $2$ .

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Step 10.2.3.1

Cancel the common factor.

$- 4 (\frac{2 \cdot 0^{2 (\frac{3}{2})}}{3} - \frac{2 {(a^{2})}^{\frac{3}{2}}}{3})$

Step 10.2.3.2

Rewrite the expression.

$- 4 (\frac{2 \cdot 0^{3}}{3} - \frac{2 {(a^{2})}^{\frac{3}{2}}}{3})$

Step 10.2.4

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

$- 4 (\frac{2 \cdot 0}{3} - \frac{2 {(a^{2})}^{\frac{3}{2}}}{3})$

Step 10.2.5

Multiply $2$ by $0$ .

$- 4 (\frac{0}{3} - \frac{2 {(a^{2})}^{\frac{3}{2}}}{3})$

Step 10.2.6

Cancel the common factor of $0$ and $3$ .

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Step 10.2.6.1

Factor $3$ out of $0$ .

$- 4 (\frac{3 (0)}{3} - \frac{2 {(a^{2})}^{\frac{3}{2}}}{3})$

Step 10.2.6.2

Cancel the common factors.

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Step 10.2.6.2.1

Factor $3$ out of $3$ .

$- 4 (\frac{3 \cdot 0}{3 \cdot 1} - \frac{2 {(a^{2})}^{\frac{3}{2}}}{3})$

Step 10.2.6.2.2

Cancel the common factor.

$- 4 (\frac{3 \cdot 0}{3 \cdot 1} - \frac{2 {(a^{2})}^{\frac{3}{2}}}{3})$

Step 10.2.6.2.3

Rewrite the expression.

$- 4 (\frac{0}{1} - \frac{2 {(a^{2})}^{\frac{3}{2}}}{3})$

Step 10.2.6.2.4

Divide $0$ by $1$ .

$- 4 (0 - \frac{2 {(a^{2})}^{\frac{3}{2}}}{3})$

Step 10.2.7

Multiply the exponents in ${(a^{2})}^{\frac{3}{2}}$ .

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Step 10.2.7.1

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

$- 4 (0 - \frac{2 a^{2 (\frac{3}{2})}}{3})$

Step 10.2.7.2

Cancel the common factor of $2$ .

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Step 10.2.7.2.1

Cancel the common factor.

$- 4 (0 - \frac{2 a^{2 (\frac{3}{2})}}{3})$

Step 10.2.7.2.2

Rewrite the expression.

$- 4 (0 - \frac{2 a^{3}}{3})$

Step 10.2.8

Subtract $\frac{2 a^{3}}{3}$ from $0$ .

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

Step 10.2.9

Multiply $- 1$ by $- 4$ .

$4 \frac{2 a^{3}}{3}$

Step 10.2.10

Combine $4$ and $\frac{2 a^{3}}{3}$ .

$\frac{4 (2 a^{3})}{3}$

Step 10.2.11

Multiply $2$ by $4$ .

$\frac{8 a^{3}}{3}$

Step 11

Reorder terms.

$\frac{8}{3} a^{3}$

Step 12

Combine $\frac{8}{3}$ and $a^{3}$ .

$\frac{8 a^{3}}{3}$