Calculus Examples

$\int_{0}^{a^{\frac{1}{2}}} x^{3} \sqrt{a^{2} - x^{4}} d x$

Step 1

Let $u = a^{2} - x^{4}$ . Then $d u = - 4 x^{3} d x$ , so $- \frac{1}{4} d u = x^{3} d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 1.1

Let $u = a^{2} - x^{4}$ . Find $\frac{d u}{d x}$ .

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Step 1.1.1

Differentiate $a^{2} - x^{4}$ .

$\frac{d}{d x} [a^{2} - x^{4}]$

Step 1.1.2

Differentiate.

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Step 1.1.2.1

By the Sum Rule, the derivative of $a^{2} - x^{4}$ with respect to $x$ is $\frac{d}{d x} [a^{2}] + \frac{d}{d x} [- x^{4}]$ .

$\frac{d}{d x} [a^{2}] + \frac{d}{d x} [- x^{4}]$

Step 1.1.2.2

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

$0 + \frac{d}{d x} [- x^{4}]$

Step 1.1.3

Evaluate $\frac{d}{d x} [- x^{4}]$ .

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Step 1.1.3.1

Since $- 1$ is constant with respect to $x$ , the derivative of $- x^{4}$ with respect to $x$ is $- \frac{d}{d x} [x^{4}]$ .

$0 - \frac{d}{d x} [x^{4}]$

Step 1.1.3.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 4$ .

$0 - (4 x^{3})$

Step 1.1.3.3

Multiply $4$ by $- 1$ .

$0 - 4 x^{3}$

Step 1.1.4

Subtract $4 x^{3}$ from $0$ .

$- 4 x^{3}$

Step 1.2

Substitute the lower limit in for $x$ in $u = a^{2} - x^{4}$ .

$u_{lower} = a^{2} - 0^{4}$

Step 1.3

Simplify.

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Step 1.3.1

Simplify each term.

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Step 1.3.1.1

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

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

Step 1.3.1.2

Multiply $- 1$ by $0$ .

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

Step 1.3.2

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

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

Step 1.4

Substitute the upper limit in for $x$ in $u = a^{2} - x^{4}$ .

$u_{upper} = a^{2} - {(a^{\frac{1}{2}})}^{4}$

Step 1.5

Simplify.

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Step 1.5.1

Multiply the exponents in ${(a^{\frac{1}{2}})}^{4}$ .

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Step 1.5.1.1

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

$u_{upper} = a^{2} - a^{\frac{1}{2} \cdot 4}$

Step 1.5.1.2

Cancel the common factor of $2$ .

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Step 1.5.1.2.1

Factor $2$ out of $4$ .

$u_{upper} = a^{2} - a^{\frac{1}{2} \cdot (2 (2))}$

Step 1.5.1.2.2

Cancel the common factor.

$u_{upper} = a^{2} - a^{\frac{1}{2} \cdot (2 \cdot 2)}$

Step 1.5.1.2.3

Rewrite the expression.

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

Step 1.5.2

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

$u_{upper} = 0$

Step 1.6

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 1.7

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

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

Step 2

Simplify.

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Step 2.1

Move the negative in front of the fraction.

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

Step 2.2

Combine $\sqrt{u}$ and $\frac{1}{4}$ .

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

Step 3

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

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

Step 4

Since $\frac{1}{4}$ is constant with respect to $u$ , move $\frac{1}{4}$ out of the integral.

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

Step 5

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

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

Step 6

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

$- \frac{1}{4} \frac{2}{3} u^{\frac{3}{2}}]_{a^{2}}^{0}$

Step 7

Substitute and simplify.

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Step 7.1

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

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

Step 7.2

Simplify.

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Step 7.2.1

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

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

Step 7.2.2

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

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

Step 7.2.3

Cancel the common factor of $2$ .

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Step 7.2.3.1

Cancel the common factor.

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

Step 7.2.3.2

Rewrite the expression.

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

Step 7.2.4

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

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

Step 7.2.5

Multiply $\frac{2}{3}$ by $0$ .

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

Step 7.2.6

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

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Step 7.2.6.1

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

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

Step 7.2.6.2

Cancel the common factor of $2$ .

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Step 7.2.6.2.1

Cancel the common factor.

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

Step 7.2.6.2.2

Rewrite the expression.

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

Step 7.2.7

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

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

Step 7.2.8

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

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

Step 7.2.9

Multiply $- 1$ by $- 1$ .

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

Step 7.2.10

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

$\frac{1}{4} \cdot \frac{a^{3} \cdot 2}{3}$

Step 7.2.11

Multiply $\frac{1}{4}$ by $\frac{a^{3} \cdot 2}{3}$ .

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

Step 7.2.12

Multiply $4$ by $3$ .

$\frac{a^{3} \cdot 2}{12}$

Step 7.2.13

Move $2$ to the left of $a^{3}$ .

$\frac{2 \cdot a^{3}}{12}$

Step 7.2.14

Cancel the common factor of $2$ and $12$ .

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Step 7.2.14.1

Factor $2$ out of $2 a^{3}$ .

$\frac{2 (a^{3})}{12}$

Step 7.2.14.2

Cancel the common factors.

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Step 7.2.14.2.1

Factor $2$ out of $12$ .

$\frac{2 a^{3}}{2 \cdot 6}$

Step 7.2.14.2.2

Cancel the common factor.

$\frac{2 a^{3}}{2 \cdot 6}$

Step 7.2.14.2.3

Rewrite the expression.

$\frac{a^{3}}{6}$

Step 8

Reorder terms.

$\frac{1}{6} a^{3}$

Step 9

Combine $\frac{1}{6}$ and $a^{3}$ .

$\frac{a^{3}}{6}$