Calculus Examples

$\int_{- 6}^{6} 36 - x^{2} d x$

Step 1

Split the single integral into multiple integrals.

$\int_{- 6}^{6} 36 d x + \int_{- 6}^{6} - x^{2} d x$

Step 2

Apply the constant rule.

$36 x]_{- 6}^{6} + \int_{- 6}^{6} - x^{2} d x$

Step 3

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

$36 x]_{- 6}^{6} - \int_{- 6}^{6} x^{2} d x$

Step 4

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

$36 x]_{- 6}^{6} - (\frac{1}{3} x^{3}]_{- 6}^{6})$

Step 5

Simplify the answer.

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Step 5.1

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

$36 x]_{- 6}^{6} - (\frac{x^{3}}{3}]_{- 6}^{6})$

Step 5.2

Substitute and simplify.

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Step 5.2.1

Evaluate $36 x$ at $6$ and at $- 6$ .

$(36 \cdot 6) - 36 \cdot - 6 - (\frac{x^{3}}{3}]_{- 6}^{6})$

Step 5.2.2

Evaluate $\frac{x^{3}}{3}$ at $6$ and at $- 6$ .

$36 \cdot 6 - 36 \cdot - 6 - (\frac{6^{3}}{3} - \frac{{(- 6)}^{3}}{3})$

Step 5.2.3

Simplify.

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Step 5.2.3.1

Multiply $36$ by $6$ .

$216 - 36 \cdot - 6 - (\frac{6^{3}}{3} - \frac{{(- 6)}^{3}}{3})$

Step 5.2.3.2

Multiply $- 36$ by $- 6$ .

$216 + 216 - (\frac{6^{3}}{3} - \frac{{(- 6)}^{3}}{3})$

Step 5.2.3.3

Add $216$ and $216$ .

$432 - (\frac{6^{3}}{3} - \frac{{(- 6)}^{3}}{3})$

Step 5.2.3.4

Raise $6$ to the power of $3$ .

$432 - (\frac{216}{3} - \frac{{(- 6)}^{3}}{3})$

Step 5.2.3.5

Cancel the common factor of $216$ and $3$ .

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Step 5.2.3.5.1

Factor $3$ out of $216$ .

$432 - (\frac{3 \cdot 72}{3} - \frac{{(- 6)}^{3}}{3})$

Step 5.2.3.5.2

Cancel the common factors.

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Step 5.2.3.5.2.1

Factor $3$ out of $3$ .

$432 - (\frac{3 \cdot 72}{3 (1)} - \frac{{(- 6)}^{3}}{3})$

Step 5.2.3.5.2.2

Cancel the common factor.

$432 - (\frac{3 \cdot 72}{3 \cdot 1} - \frac{{(- 6)}^{3}}{3})$

Step 5.2.3.5.2.3

Rewrite the expression.

$432 - (\frac{72}{1} - \frac{{(- 6)}^{3}}{3})$

Step 5.2.3.5.2.4

Divide $72$ by $1$ .

$432 - (72 - \frac{{(- 6)}^{3}}{3})$

Step 5.2.3.6

Raise $- 6$ to the power of $3$ .

$432 - (72 - \frac{- 216}{3})$

Step 5.2.3.7

Cancel the common factor of $- 216$ and $3$ .

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Step 5.2.3.7.1

Factor $3$ out of $- 216$ .

$432 - (72 - \frac{3 \cdot - 72}{3})$

Step 5.2.3.7.2

Cancel the common factors.

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Step 5.2.3.7.2.1

Factor $3$ out of $3$ .

$432 - (72 - \frac{3 \cdot - 72}{3 (1)})$

Step 5.2.3.7.2.2

Cancel the common factor.

$432 - (72 - \frac{3 \cdot - 72}{3 \cdot 1})$

Step 5.2.3.7.2.3

Rewrite the expression.

$432 - (72 - \frac{- 72}{1})$

Step 5.2.3.7.2.4

Divide $- 72$ by $1$ .

$432 - (72 - - 72)$

Step 5.2.3.8

Multiply $- 1$ by $- 72$ .

$432 - (72 + 72)$

Step 5.2.3.9

Add $72$ and $72$ .

$432 - 1 \cdot 144$

Step 5.2.3.10

Multiply $- 1$ by $144$ .

$432 - 144$

Step 5.2.3.11

Subtract $144$ from $432$ .

$288$

Step 6