Calculus Examples

$\int_{0}^{1} b \sqrt{1 - b^{2}} d b$

Step 1

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

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Step 1.1

Let $u = 1 - b^{2}$ . Find $\frac{d u}{d b}$ .

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Step 1.1.1

Differentiate $1 - b^{2}$ .

$\frac{d}{d b} [1 - b^{2}]$

Step 1.1.2

Differentiate.

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Step 1.1.2.1

By the Sum Rule, the derivative of $1 - b^{2}$ with respect to $b$ is $\frac{d}{d b} [1] + \frac{d}{d b} [- b^{2}]$ .

$\frac{d}{d b} [1] + \frac{d}{d b} [- b^{2}]$

Step 1.1.2.2

Since $1$ is constant with respect to $b$ , the derivative of $1$ with respect to $b$ is $0$ .

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

Step 1.1.3

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

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Step 1.1.3.1

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

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

Step 1.1.3.2

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

$0 - (2 b)$

Step 1.1.3.3

Multiply $2$ by $- 1$ .

$0 - 2 b$

Step 1.1.4

Subtract $2 b$ from $0$ .

$- 2 b$

Step 1.2

Substitute the lower limit in for $b$ in $u = 1 - b^{2}$ .

$u_{lower} = 1 - 0^{2}$

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 $0$ .

$u_{lower} = 1 - 0$

Step 1.3.1.2

Multiply $- 1$ by $0$ .

$u_{lower} = 1 + 0$

Step 1.3.2

Add $1$ and $0$ .

$u_{lower} = 1$

Step 1.4

Substitute the upper limit in for $b$ in $u = 1 - b^{2}$ .

$u_{upper} = 1 - 1^{2}$

Step 1.5

Simplify.

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Step 1.5.1

Simplify each term.

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Step 1.5.1.1

One to any power is one.

$u_{upper} = 1 - 1 \cdot 1$

Step 1.5.1.2

Multiply $- 1$ by $1$ .

$u_{upper} = 1 - 1$

Step 1.5.2

Subtract $1$ from $1$ .

$u_{upper} = 0$

Step 1.6

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

$u_{lower} = 1$

$u_{upper} = 0$

Step 1.7

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

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

Step 2

Simplify.

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Step 2.1

Move the negative in front of the fraction.

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

Step 2.2

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

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

Step 3

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

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

Step 4

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

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

Step 5

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

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

Step 6

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

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

Step 7

Substitute and simplify.

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Step 7.1

Evaluate $\frac{2}{3} u^{\frac{3}{2}}$ at $0$ and at $1$ .

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

Step 7.2

Simplify.

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Step 7.2.1

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

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

Step 7.2.2

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

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

Step 7.2.3

Cancel the common factor of $2$ .

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Step 7.2.3.1

Cancel the common factor.

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

Step 7.2.3.2

Rewrite the expression.

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

Step 7.2.4

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

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

Step 7.2.5

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

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

Step 7.2.6

One to any power is one.

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

Step 7.2.7

Multiply $- 1$ by $1$ .

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

Step 7.2.8

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

$- \frac{1}{2} (- \frac{2}{3})$

Step 7.2.9

Multiply $- 1$ by $- 1$ .

$1 (\frac{1}{2}) \frac{2}{3}$

Step 7.2.10

Multiply $\frac{1}{2}$ by $1$ .

$\frac{1}{2} \cdot \frac{2}{3}$

Step 7.2.11

Multiply $\frac{1}{2}$ by $\frac{2}{3}$ .

$\frac{2}{2 \cdot 3}$

Step 7.2.12

Multiply $2$ by $3$ .

$\frac{2}{6}$

Step 7.2.13

Cancel the common factor of $2$ and $6$ .

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Step 7.2.13.1

Factor $2$ out of $2$ .

$\frac{2 (1)}{6}$

Step 7.2.13.2

Cancel the common factors.

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Step 7.2.13.2.1

Factor $2$ out of $6$ .

$\frac{2 \cdot 1}{2 \cdot 3}$

Step 7.2.13.2.2

Cancel the common factor.

$\frac{2 \cdot 1}{2 \cdot 3}$

Step 7.2.13.2.3

Rewrite the expression.

$\frac{1}{3}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$\frac{1}{3}$

Decimal Form:

$0.\overline{3}$

Step 9