Calculus Examples

$\int_{- 3}^{3} [(18 - x^{2}) - (x^{2})] d x$

Step 1

Remove parentheses.

$\int_{- 3}^{3} (18 - x^{2}) - (x^{2}) d x$

Step 2

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

$\int_{- 3}^{3} 18 - 2 x^{2} d x$

Step 3

Split the single integral into multiple integrals.

$\int_{- 3}^{3} 18 d x + \int_{- 3}^{3} - 2 x^{2} d x$

Step 4

Apply the constant rule.

$18 x]_{- 3}^{3} + \int_{- 3}^{3} - 2 x^{2} d x$

Step 5

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

$18 x]_{- 3}^{3} - 2 \int_{- 3}^{3} x^{2} d x$

Step 6

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

$18 x]_{- 3}^{3} - 2 (\frac{1}{3} x^{3}]_{- 3}^{3})$

Step 7

Simplify the answer.

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Step 7.1

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

$18 x]_{- 3}^{3} - 2 (\frac{x^{3}}{3}]_{- 3}^{3})$

Step 7.2

Substitute and simplify.

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Step 7.2.1

Evaluate $18 x$ at $3$ and at $- 3$ .

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

Step 7.2.2

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

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

Step 7.2.3

Simplify.

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Step 7.2.3.1

Multiply $18$ by $3$ .

$54 - 18 \cdot - 3 - 2 (\frac{3^{3}}{3} - \frac{{(- 3)}^{3}}{3})$

Step 7.2.3.2

Multiply $- 18$ by $- 3$ .

$54 + 54 - 2 (\frac{3^{3}}{3} - \frac{{(- 3)}^{3}}{3})$

Step 7.2.3.3

Add $54$ and $54$ .

$108 - 2 (\frac{3^{3}}{3} - \frac{{(- 3)}^{3}}{3})$

Step 7.2.3.4

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

$108 - 2 (\frac{27}{3} - \frac{{(- 3)}^{3}}{3})$

Step 7.2.3.5

Cancel the common factor of $27$ and $3$ .

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Step 7.2.3.5.1

Factor $3$ out of $27$ .

$108 - 2 (\frac{3 \cdot 9}{3} - \frac{{(- 3)}^{3}}{3})$

Step 7.2.3.5.2

Cancel the common factors.

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Step 7.2.3.5.2.1

Factor $3$ out of $3$ .

$108 - 2 (\frac{3 \cdot 9}{3 (1)} - \frac{{(- 3)}^{3}}{3})$

Step 7.2.3.5.2.2

Cancel the common factor.

$108 - 2 (\frac{3 \cdot 9}{3 \cdot 1} - \frac{{(- 3)}^{3}}{3})$

Step 7.2.3.5.2.3

Rewrite the expression.

$108 - 2 (\frac{9}{1} - \frac{{(- 3)}^{3}}{3})$

Step 7.2.3.5.2.4

Divide $9$ by $1$ .

$108 - 2 (9 - \frac{{(- 3)}^{3}}{3})$

Step 7.2.3.6

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

$108 - 2 (9 - \frac{- 27}{3})$

Step 7.2.3.7

Cancel the common factor of $- 27$ and $3$ .

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Step 7.2.3.7.1

Factor $3$ out of $- 27$ .

$108 - 2 (9 - \frac{3 \cdot - 9}{3})$

Step 7.2.3.7.2

Cancel the common factors.

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Step 7.2.3.7.2.1

Factor $3$ out of $3$ .

$108 - 2 (9 - \frac{3 \cdot - 9}{3 (1)})$

Step 7.2.3.7.2.2

Cancel the common factor.

$108 - 2 (9 - \frac{3 \cdot - 9}{3 \cdot 1})$

Step 7.2.3.7.2.3

Rewrite the expression.

$108 - 2 (9 - \frac{- 9}{1})$

Step 7.2.3.7.2.4

Divide $- 9$ by $1$ .

$108 - 2 (9 - - 9)$

Step 7.2.3.8

Multiply $- 1$ by $- 9$ .

$108 - 2 (9 + 9)$

Step 7.2.3.9

Add $9$ and $9$ .

$108 - 2 \cdot 18$

Step 7.2.3.10

Multiply $- 2$ by $18$ .

$108 - 36$

Step 7.2.3.11

Subtract $36$ from $108$ .

$72$

Step 8