Calculus Examples

$\int_{0}^{7} \sqrt{y + 9} d y$

Step 1

Let $u = y + 9$ . Then $d u = d y$ . Rewrite using $u$ and $d$ $u$ .

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Step 1.1

Let $u = y + 9$ . Find $\frac{d u}{d y}$ .

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Step 1.1.1

Differentiate $y + 9$ .

$\frac{d}{d y} [y + 9]$

Step 1.1.2

By the Sum Rule, the derivative of $y + 9$ with respect to $y$ is $\frac{d}{d y} [y] + \frac{d}{d y} [9]$ .

$\frac{d}{d y} [y] + \frac{d}{d y} [9]$

Step 1.1.3

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

$1 + \frac{d}{d y} [9]$

Step 1.1.4

Since $9$ is constant with respect to $y$ , the derivative of $9$ with respect to $y$ is $0$ .

$1 + 0$

Step 1.1.5

Add $1$ and $0$ .

$1$

Step 1.2

Substitute the lower limit in for $y$ in $u = y + 9$ .

$u_{lower} = 0 + 9$

Step 1.3

Add $0$ and $9$ .

$u_{lower} = 9$

Step 1.4

Substitute the upper limit in for $y$ in $u = y + 9$ .

$u_{upper} = 7 + 9$

Step 1.5

Add $7$ and $9$ .

$u_{upper} = 16$

Step 1.6

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

$u_{lower} = 9$

$u_{upper} = 16$

Step 1.7

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

$\int_{9}^{16} \sqrt{u} d u$

Step 2

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

$\int_{9}^{16} u^{\frac{1}{2}} d u$

Step 3

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

$\frac{2}{3} u^{\frac{3}{2}}]_{9}^{16}$

Step 4

Substitute and simplify.

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Step 4.1

Evaluate $\frac{2}{3} u^{\frac{3}{2}}$ at $16$ and at $9$ .

$(\frac{2}{3} \cdot 16^{\frac{3}{2}}) - \frac{2}{3} \cdot 9^{\frac{3}{2}}$

Step 4.2

Simplify.

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Step 4.2.1

Rewrite $16$ as $4^{2}$ .

$\frac{2}{3} \cdot {(4^{2})}^{\frac{3}{2}} - \frac{2}{3} \cdot 9^{\frac{3}{2}}$

Step 4.2.2

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

$\frac{2}{3} \cdot 4^{2 (\frac{3}{2})} - \frac{2}{3} \cdot 9^{\frac{3}{2}}$

Step 4.2.3

Cancel the common factor of $2$ .

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Step 4.2.3.1

Cancel the common factor.

$\frac{2}{3} \cdot 4^{2 (\frac{3}{2})} - \frac{2}{3} \cdot 9^{\frac{3}{2}}$

Step 4.2.3.2

Rewrite the expression.

$\frac{2}{3} \cdot 4^{3} - \frac{2}{3} \cdot 9^{\frac{3}{2}}$

Step 4.2.4

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

$\frac{2}{3} \cdot 64 - \frac{2}{3} \cdot 9^{\frac{3}{2}}$

Step 4.2.5

Combine $\frac{2}{3}$ and $64$ .

$\frac{2 \cdot 64}{3} - \frac{2}{3} \cdot 9^{\frac{3}{2}}$

Step 4.2.6

Multiply $2$ by $64$ .

$\frac{128}{3} - \frac{2}{3} \cdot 9^{\frac{3}{2}}$

Step 4.2.7

Rewrite $9$ as $3^{2}$ .

$\frac{128}{3} - \frac{2}{3} \cdot {(3^{2})}^{\frac{3}{2}}$

Step 4.2.8

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

$\frac{128}{3} - \frac{2}{3} \cdot 3^{2 (\frac{3}{2})}$

Step 4.2.9

Cancel the common factor of $2$ .

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Step 4.2.9.1

Cancel the common factor.

$\frac{128}{3} - \frac{2}{3} \cdot 3^{2 (\frac{3}{2})}$

Step 4.2.9.2

Rewrite the expression.

$\frac{128}{3} - \frac{2}{3} \cdot 3^{3}$

Step 4.2.10

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

$\frac{128}{3} - \frac{2}{3} \cdot 27$

Step 4.2.11

Multiply $27$ by $- 1$ .

$\frac{128}{3} - 27 (\frac{2}{3})$

Step 4.2.12

Combine $- 27$ and $\frac{2}{3}$ .

$\frac{128}{3} + \frac{- 27 \cdot 2}{3}$

Step 4.2.13

Multiply $- 27$ by $2$ .

$\frac{128}{3} + \frac{- 54}{3}$

Step 4.2.14

Cancel the common factor of $- 54$ and $3$ .

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Step 4.2.14.1

Factor $3$ out of $- 54$ .

$\frac{128}{3} + \frac{3 \cdot - 18}{3}$

Step 4.2.14.2

Cancel the common factors.

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Step 4.2.14.2.1

Factor $3$ out of $3$ .

$\frac{128}{3} + \frac{3 \cdot - 18}{3 (1)}$

Step 4.2.14.2.2

Cancel the common factor.

$\frac{128}{3} + \frac{3 \cdot - 18}{3 \cdot 1}$

Step 4.2.14.2.3

Rewrite the expression.

$\frac{128}{3} + \frac{- 18}{1}$

Step 4.2.14.2.4

Divide $- 18$ by $1$ .

$\frac{128}{3} - 18$

Step 4.2.15

To write $- 18$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{128}{3} - 18 \cdot \frac{3}{3}$

Step 4.2.16

Combine $- 18$ and $\frac{3}{3}$ .

$\frac{128}{3} + \frac{- 18 \cdot 3}{3}$

Step 4.2.17

Combine the numerators over the common denominator.

$\frac{128 - 18 \cdot 3}{3}$

Step 4.2.18

Simplify the numerator.

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Step 4.2.18.1

Multiply $- 18$ by $3$ .

$\frac{128 - 54}{3}$

Step 4.2.18.2

Subtract $54$ from $128$ .

$\frac{74}{3}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$\frac{74}{3}$

Decimal Form:

$24.\overline{6}$

Mixed Number Form:

$24 \frac{2}{3}$

Step 6