Calculus Examples

$\int_{- 4}^{4} 16 - x^{2} d x$

Step 1

Split the single integral into multiple integrals.

$\int_{- 4}^{4} 16 d x + \int_{- 4}^{4} - x^{2} d x$

Step 2

Apply the constant rule.

$16 x]_{- 4}^{4} + \int_{- 4}^{4} - x^{2} d x$

Step 3

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

$16 x]_{- 4}^{4} - \int_{- 4}^{4} x^{2} d x$

Step 4

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

$16 x]_{- 4}^{4} - (\frac{1}{3} x^{3}]_{- 4}^{4})$

Step 5

Simplify the answer.

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Step 5.1

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

$16 x]_{- 4}^{4} - (\frac{x^{3}}{3}]_{- 4}^{4})$

Step 5.2

Substitute and simplify.

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Step 5.2.1

Evaluate $16 x$ at $4$ and at $- 4$ .

$(16 \cdot 4) - 16 \cdot - 4 - (\frac{x^{3}}{3}]_{- 4}^{4})$

Step 5.2.2

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

$16 \cdot 4 - 16 \cdot - 4 - (\frac{4^{3}}{3} - \frac{{(- 4)}^{3}}{3})$

Step 5.2.3

Simplify.

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Step 5.2.3.1

Multiply $16$ by $4$ .

$64 - 16 \cdot - 4 - (\frac{4^{3}}{3} - \frac{{(- 4)}^{3}}{3})$

Step 5.2.3.2

Multiply $- 16$ by $- 4$ .

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

Step 5.2.3.3

Add $64$ and $64$ .

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

Step 5.2.3.4

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

$128 - (\frac{64}{3} - \frac{{(- 4)}^{3}}{3})$

Step 5.2.3.5

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

$128 - (\frac{64}{3} - \frac{- 64}{3})$

Step 5.2.3.6

Move the negative in front of the fraction.

$128 - (\frac{64}{3} - - \frac{64}{3})$

Step 5.2.3.7

Multiply $- 1$ by $- 1$ .

$128 - (\frac{64}{3} + 1 (\frac{64}{3}))$

Step 5.2.3.8

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

$128 - (\frac{64}{3} + \frac{64}{3})$

Step 5.2.3.9

Combine the numerators over the common denominator.

$128 - \frac{64 + 64}{3}$

Step 5.2.3.10

Add $64$ and $64$ .

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

Step 5.2.3.11

To write $128$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

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

Step 5.2.3.12

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

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

Step 5.2.3.13

Combine the numerators over the common denominator.

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

Step 5.2.3.14

Simplify the numerator.

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Step 5.2.3.14.1

Multiply $128$ by $3$ .

$\frac{384 - 128}{3}$

Step 5.2.3.14.2

Subtract $128$ from $384$ .

$\frac{256}{3}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$\frac{256}{3}$

Decimal Form:

$85.\overline{3}$

Mixed Number Form:

$85 \frac{1}{3}$

Step 7