Calculus Examples

$\int_{- 4}^{4} [(32 - x^{2}) - (x^{2})] d x$

Step 1

Remove parentheses.

$\int_{- 4}^{4} (32 - x^{2}) - (x^{2}) d x$

Step 2

Subtract $x^{2}$ from $- x^{2}$ .

$\int_{- 4}^{4} 32 - 2 x^{2} d x$

Step 3

Split the single integral into multiple integrals.

$\int_{- 4}^{4} 32 d x + \int_{- 4}^{4} - 2 x^{2} d x$

Step 4

Apply the constant rule.

$32 x]_{- 4}^{4} + \int_{- 4}^{4} - 2 x^{2} d x$

Step 5

Since $- 2$ is constant with respect to $x$ , move $- 2$ out of the integral.

$32 x]_{- 4}^{4} - 2 \int_{- 4}^{4} x^{2} d x$

Step 6

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

$32 x]_{- 4}^{4} - 2 (\frac{1}{3} x^{3}]_{- 4}^{4})$

Step 7

Simplify the answer.

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Step 7.1

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

$32 x]_{- 4}^{4} - 2 (\frac{x^{3}}{3}]_{- 4}^{4})$

Step 7.2

Substitute and simplify.

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Step 7.2.1

Evaluate $32 x$ at $4$ and at $- 4$ .

$(32 \cdot 4) - 32 \cdot - 4 - 2 (\frac{x^{3}}{3}]_{- 4}^{4})$

Step 7.2.2

Evaluate $\frac{x^{3}}{3}$ at $4$ and at $- 4$ .

$32 \cdot 4 - 32 \cdot - 4 - 2 (\frac{4^{3}}{3} - \frac{{(- 4)}^{3}}{3})$

Step 7.2.3

Simplify.

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Step 7.2.3.1

Multiply $32$ by $4$ .

$128 - 32 \cdot - 4 - 2 (\frac{4^{3}}{3} - \frac{{(- 4)}^{3}}{3})$

Step 7.2.3.2

Multiply $- 32$ by $- 4$ .

$128 + 128 - 2 (\frac{4^{3}}{3} - \frac{{(- 4)}^{3}}{3})$

Step 7.2.3.3

Add $128$ and $128$ .

$256 - 2 (\frac{4^{3}}{3} - \frac{{(- 4)}^{3}}{3})$

Step 7.2.3.4

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

$256 - 2 (\frac{64}{3} - \frac{{(- 4)}^{3}}{3})$

Step 7.2.3.5

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

$256 - 2 (\frac{64}{3} - \frac{- 64}{3})$

Step 7.2.3.6

Move the negative in front of the fraction.

$256 - 2 (\frac{64}{3} - - \frac{64}{3})$

Step 7.2.3.7

Multiply $- 1$ by $- 1$ .

$256 - 2 (\frac{64}{3} + 1 (\frac{64}{3}))$

Step 7.2.3.8

Multiply $\frac{64}{3}$ by $1$ .

$256 - 2 (\frac{64}{3} + \frac{64}{3})$

Step 7.2.3.9

Combine the numerators over the common denominator.

$256 - 2 \frac{64 + 64}{3}$

Step 7.2.3.10

Add $64$ and $64$ .

$256 - 2 (\frac{128}{3})$

Step 7.2.3.11

Combine $- 2$ and $\frac{128}{3}$ .

$256 + \frac{- 2 \cdot 128}{3}$

Step 7.2.3.12

Multiply $- 2$ by $128$ .

$256 + \frac{- 256}{3}$

Step 7.2.3.13

Move the negative in front of the fraction.

$256 - \frac{256}{3}$

Step 7.2.3.14

To write $256$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$256 \cdot \frac{3}{3} - \frac{256}{3}$

Step 7.2.3.15

Combine $256$ and $\frac{3}{3}$ .

$\frac{256 \cdot 3}{3} - \frac{256}{3}$

Step 7.2.3.16

Combine the numerators over the common denominator.

$\frac{256 \cdot 3 - 256}{3}$

Step 7.2.3.17

Simplify the numerator.

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Step 7.2.3.17.1

Multiply $256$ by $3$ .

$\frac{768 - 256}{3}$

Step 7.2.3.17.2

Subtract $256$ from $768$ .

$\frac{512}{3}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$\frac{512}{3}$

Decimal Form:

$170.\overline{6}$

Mixed Number Form:

$170 \frac{2}{3}$

Step 9