Calculus Examples

$\int_{0}^{5} \sqrt{y + 4} d y$

Step 1

Let $u = y + 4$ . Then $d u = d y$ . Rewrite using $u$ and $d$ $u$ .

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Step 1.1

Let $u = y + 4$ . Find $\frac{d u}{d y}$ .

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Step 1.1.1

Differentiate $y + 4$ .

$\frac{d}{d y} [y + 4]$

Step 1.1.2

By the Sum Rule, the derivative of $y + 4$ with respect to $y$ is $\frac{d}{d y} [y] + \frac{d}{d y} [4]$ .

$\frac{d}{d y} [y] + \frac{d}{d y} [4]$

Step 1.1.3

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

$1 + \frac{d}{d y} [4]$

Step 1.1.4

Since $4$ is constant with respect to $y$ , the derivative of $4$ 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 $u = y + 4$ .

$u_{lower} = 0 + 4$

Step 1.3

Add $0$ and $4$ .

$u_{lower} = 4$

Step 1.4

Substitute the upper limit in for $y$ in $u = y + 4$ .

$u_{upper} = 5 + 4$

Step 1.5

Add $5$ and $4$ .

$u_{upper} = 9$

Step 1.6

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

$u_{lower} = 4$

$u_{upper} = 9$

Step 1.7

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

$\int_{4}^{9} \sqrt{u} d u$

Step 2

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

$\int_{4}^{9} u^{\frac{1}{2}} d u$

Step 3

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

$\frac{2}{3} u^{\frac{3}{2}}]_{4}^{9}$

Step 4

Simplify the expression.

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Step 4.1

Evaluate $\frac{2}{3} u^{\frac{3}{2}}$ at $9$ and at $4$ .

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

Step 4.2

Simplify.

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Step 4.2.1

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

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

Step 4.2.2

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

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

Step 4.2.3

Cancel the common factor of $2$ .

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Step 4.2.3.1

Cancel the common factor.

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

Step 4.2.3.2

Rewrite the expression.

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

Step 4.2.4

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

$\frac{2}{3} \cdot 27 - \frac{2}{3} \cdot 4^{\frac{3}{2}}$

Step 4.3

Simplify.

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Step 4.3.1

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

$\frac{2 \cdot 27}{3} - \frac{2}{3} \cdot 4^{\frac{3}{2}}$

Step 4.3.2

Multiply $2$ by $27$ .

$\frac{54}{3} - \frac{2}{3} \cdot 4^{\frac{3}{2}}$

Step 4.3.3

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

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Step 4.3.3.1

Factor $3$ out of $54$ .

$\frac{3 \cdot 18}{3} - \frac{2}{3} \cdot 4^{\frac{3}{2}}$

Step 4.3.3.2

Cancel the common factors.

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Step 4.3.3.2.1

Factor $3$ out of $3$ .

$\frac{3 \cdot 18}{3 (1)} - \frac{2}{3} \cdot 4^{\frac{3}{2}}$

Step 4.3.3.2.2

Cancel the common factor.

$\frac{3 \cdot 18}{3 \cdot 1} - \frac{2}{3} \cdot 4^{\frac{3}{2}}$

Step 4.3.3.2.3

Rewrite the expression.

$\frac{18}{1} - \frac{2}{3} \cdot 4^{\frac{3}{2}}$

Step 4.3.3.2.4

Divide $18$ by $1$ .

$18 - \frac{2}{3} \cdot 4^{\frac{3}{2}}$

Step 4.4

Simplify.

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Step 4.4.1

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

$18 - \frac{2}{3} \cdot {(2^{2})}^{\frac{3}{2}}$

Step 4.4.2

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

$18 - \frac{2}{3} \cdot 2^{2 (\frac{3}{2})}$

Step 4.4.3

Cancel the common factor of $2$ .

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Step 4.4.3.1

Cancel the common factor.

$18 - \frac{2}{3} \cdot 2^{2 (\frac{3}{2})}$

Step 4.4.3.2

Rewrite the expression.

$18 - \frac{2}{3} \cdot 2^{3}$

Step 4.4.4

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

$18 - \frac{2}{3} \cdot 8$

Step 4.5

Simplify.

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Step 4.5.1

Multiply $8$ by $- 1$ .

$18 - 8 (\frac{2}{3})$

Step 4.5.2

Combine $- 8$ and $\frac{2}{3}$ .

$18 + \frac{- 8 \cdot 2}{3}$

Step 4.5.3

Multiply $- 8$ by $2$ .

$18 + \frac{- 16}{3}$

Step 4.5.4

Move the negative in front of the fraction.

$18 - \frac{16}{3}$

Step 4.6

Simplify.

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Step 4.6.1

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

$18 \cdot \frac{3}{3} - \frac{16}{3}$

Step 4.6.2

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

$\frac{18 \cdot 3}{3} - \frac{16}{3}$

Step 4.6.3

Combine the numerators over the common denominator.

$\frac{18 \cdot 3 - 16}{3}$

Step 4.6.4

Simplify the numerator.

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Step 4.6.4.1

Multiply $18$ by $3$ .

$\frac{54 - 16}{3}$

Step 4.6.4.2

Subtract $16$ from $54$ .

$\frac{38}{3}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$\frac{38}{3}$

Decimal Form:

$12.\overline{6}$

Mixed Number Form:

$12 \frac{2}{3}$