Calculus Examples

$\int_{- \frac{b}{2}}^{\frac{b}{2}} (h - \frac{4 h}{b^{2}} x^{2}) d x$

Step 1

Remove parentheses.

$\int_{- \frac{b}{2}}^{\frac{b}{2}} h - \frac{4 h}{b^{2}} \cdot x^{2} d x$

Step 2

Simplify.

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Step 2.1

Combine $x^{2}$ and $\frac{4 h}{b^{2}}$ .

$\int_{- \frac{b}{2}}^{\frac{b}{2}} h - \frac{x^{2} (4 h)}{b^{2}} d x$

Step 2.2

Move $4$ to the left of $x^{2}$ .

$\int_{- \frac{b}{2}}^{\frac{b}{2}} h - \frac{4 x^{2} h}{b^{2}} d x$

Step 3

Split the single integral into multiple integrals.

$\int_{- \frac{b}{2}}^{\frac{b}{2}} h d x + \int_{- \frac{b}{2}}^{\frac{b}{2}} - \frac{4 x^{2} h}{b^{2}} d x$

Step 4

Apply the constant rule.

$h x]_{- \frac{b}{2}}^{\frac{b}{2}} + \int_{- \frac{b}{2}}^{\frac{b}{2}} - \frac{4 x^{2} h}{b^{2}} d x$

Step 5

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

$h x]_{- \frac{b}{2}}^{\frac{b}{2}} - \int_{- \frac{b}{2}}^{\frac{b}{2}} \frac{4 x^{2} h}{b^{2}} d x$

Step 6

Since $\frac{4 h}{b^{2}}$ is constant with respect to $x$ , move $\frac{4 h}{b^{2}}$ out of the integral.

$h x]_{- \frac{b}{2}}^{\frac{b}{2}} - (\frac{4 h}{b^{2}} \int_{- \frac{b}{2}}^{\frac{b}{2}} x^{2} d x)$

Step 7

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

$h x]_{- \frac{b}{2}}^{\frac{b}{2}} - \frac{4 h}{b^{2}} \frac{1}{3} x^{3}]_{- \frac{b}{2}}^{\frac{b}{2}}$

Step 8

Simplify the answer.

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Step 8.1

Simplify.

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Step 8.1.1

Combine $\frac{1}{3}$ and $x^{3}$ .

$h x]_{- \frac{b}{2}}^{\frac{b}{2}} - \frac{4 h}{b^{2}} \frac{x^{3}}{3}]_{- \frac{b}{2}}^{\frac{b}{2}}$

Step 8.1.2

Combine $\frac{x^{3}}{3}]_{- \frac{b}{2}}^{\frac{b}{2}}$ and $\frac{4 h}{b^{2}}$ .

$h x]_{- \frac{b}{2}}^{\frac{b}{2}} - \frac{\frac{x^{3}}{3}]_{- \frac{b}{2}}^{\frac{b}{2}} (4 h)}{b^{2}}$

Step 8.1.3

Move $4$ to the left of $\frac{x^{3}}{3}]_{- \frac{b}{2}}^{\frac{b}{2}}$ .

$h x]_{- \frac{b}{2}}^{\frac{b}{2}} - \frac{4 (\frac{x^{3}}{3}]_{- \frac{b}{2}}^{\frac{b}{2}}) h}{b^{2}}$

Step 8.2

Substitute and simplify.

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Step 8.2.1

Evaluate $h x$ at $\frac{b}{2}$ and at $- \frac{b}{2}$ .

$(h \frac{b}{2}) - h (- \frac{b}{2}) - \frac{4 (\frac{x^{3}}{3}]_{- \frac{b}{2}}^{\frac{b}{2}}) h}{b^{2}}$

Step 8.2.2

Evaluate $\frac{x^{3}}{3}$ at $\frac{b}{2}$ and at $- \frac{b}{2}$ .

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

Step 8.2.3

Simplify.

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Step 8.2.3.1

Combine $h$ and $\frac{b}{2}$ .

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

Step 8.2.3.2

Multiply $- 1$ by $- 1$ .

$\frac{h b}{2} + 1 h \frac{b}{2} - \frac{4 ((\frac{{(\frac{b}{2})}^{3}}{3}) - \frac{{(- \frac{b}{2})}^{3}}{3}) h}{b^{2}}$

Step 8.2.3.3

Multiply $h$ by $1$ .

$\frac{h b}{2} + h \frac{b}{2} - \frac{4 ((\frac{{(\frac{b}{2})}^{3}}{3}) - \frac{{(- \frac{b}{2})}^{3}}{3}) h}{b^{2}}$

Step 8.2.3.4

Combine $h$ and $\frac{b}{2}$ .

$\frac{h b}{2} + \frac{h b}{2} - \frac{4 ((\frac{{(\frac{b}{2})}^{3}}{3}) - \frac{{(- \frac{b}{2})}^{3}}{3}) h}{b^{2}}$

Step 8.2.3.5

Add $\frac{h b}{2}$ and $\frac{h b}{2}$ .

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

Step 8.2.3.6

Combine $2$ and $\frac{h b}{2}$ .

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

Step 8.2.3.7

Cancel the common factor of $2$ .

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Step 8.2.3.7.1

Cancel the common factor.

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

Step 8.2.3.7.2

Divide $h b$ by $1$ .

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

Step 8.2.3.8

Factor $- 1$ out of $- \frac{b}{2}$ .

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

Step 8.2.3.9

Apply the product rule to $- (\frac{b}{2})$ .

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

Step 8.2.3.10

Raise $- 1$ to the power of $3$ .

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

Step 8.2.3.11

Move the negative in front of the fraction.

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

Step 8.2.3.12

Multiply $- 1$ by $- 1$ .

$h b - \frac{4 (\frac{{(\frac{b}{2})}^{3}}{3} + 1 \frac{{(\frac{b}{2})}^{3}}{3}) h}{b^{2}}$

Step 8.2.3.13

Multiply $\frac{{(\frac{b}{2})}^{3}}{3}$ by $1$ .

$h b - \frac{4 (\frac{{(\frac{b}{2})}^{3}}{3} + \frac{{(\frac{b}{2})}^{3}}{3}) h}{b^{2}}$

Step 8.2.3.14

Add $\frac{{(\frac{b}{2})}^{3}}{3}$ and $\frac{{(\frac{b}{2})}^{3}}{3}$ .

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

Step 8.2.3.15

Multiply $2$ by $4$ .

$h b - \frac{8 \frac{{(\frac{b}{2})}^{3}}{3} h}{b^{2}}$

Step 8.3

Reorder terms.

$h b - \frac{8 (\frac{1}{3}) {(\frac{1}{2} b)}^{3} h}{b^{2}}$

Step 9

Simplify each term.

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Step 9.1

Simplify the numerator.

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Step 9.1.1

Combine $8$ and $\frac{1}{3}$ .

$h b - \frac{\frac{8}{3} {(\frac{1}{2} b)}^{3} h}{b^{2}}$

Step 9.1.2

Combine $\frac{8}{3}$ and ${(\frac{1}{2} b)}^{3}$ .

$h b - \frac{\frac{8 {(\frac{1}{2} b)}^{3}}{3} h}{b^{2}}$

Step 9.1.3

Combine $\frac{8 {(\frac{1}{2} b)}^{3}}{3}$ and $h$ .

$h b - \frac{\frac{8 {(\frac{1}{2} b)}^{3} h}{3}}{b^{2}}$

Step 9.2

Simplify the numerator.

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Step 9.2.1

Combine $\frac{1}{2}$ and $b$ .

$h b - \frac{\frac{8 {(\frac{b}{2})}^{3} h}{3}}{b^{2}}$

Step 9.2.2

Apply the product rule to $\frac{b}{2}$ .

$h b - \frac{\frac{8 \frac{b^{3}}{2^{3}} h}{3}}{b^{2}}$

Step 9.2.3

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

$h b - \frac{\frac{8 \frac{b^{3}}{8} h}{3}}{b^{2}}$

Step 9.2.4

Combine exponents.

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Step 9.2.4.1

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

$h b - \frac{\frac{\frac{8 b^{3}}{8} h}{3}}{b^{2}}$

Step 9.2.4.2

Combine $\frac{8 b^{3}}{8}$ and $h$ .

$h b - \frac{\frac{\frac{8 b^{3} h}{8}}{3}}{b^{2}}$

Step 9.2.5

Reduce the expression by cancelling the common factors.

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Step 9.2.5.1

Reduce the expression $\frac{8 b^{3} h}{8}$ by cancelling the common factors.

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Step 9.2.5.1.1

Cancel the common factor.

$h b - \frac{\frac{\frac{8 b^{3} h}{8}}{3}}{b^{2}}$

Step 9.2.5.1.2

Rewrite the expression.

$h b - \frac{\frac{\frac{b^{3} h}{1}}{3}}{b^{2}}$

Step 9.2.5.2

Divide $b^{3} h$ by $1$ .

$h b - \frac{\frac{b^{3} h}{3}}{b^{2}}$

Step 9.3

Multiply the numerator by the reciprocal of the denominator.

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

Step 9.4

Combine.

$h b - \frac{b^{3} h \cdot 1}{3 b^{2}}$

Step 9.5

Cancel the common factor of $b^{3}$ and $b^{2}$ .

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Step 9.5.1

Factor $b^{2}$ out of $b^{3} h \cdot 1$ .

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

Step 9.5.2

Cancel the common factors.

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Step 9.5.2.1

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

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

Step 9.5.2.2

Cancel the common factor.

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

Step 9.5.2.3

Rewrite the expression.

$h b - \frac{b h \cdot 1}{3}$

Step 9.6

Multiply $b$ by $1$ .

$h b - \frac{b h}{3}$