Calculus Examples

$\int_{- 2}^{3} 36 - x^{2} d x$

Step 1

Split the single integral into multiple integrals.

$\int_{- 2}^{3} 36 d x + \int_{- 2}^{3} - x^{2} d x$

Step 2

Apply the constant rule.

$36 x]_{- 2}^{3} + \int_{- 2}^{3} - x^{2} d x$

Step 3

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

$36 x]_{- 2}^{3} - \int_{- 2}^{3} x^{2} d x$

Step 4

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

$36 x]_{- 2}^{3} - (\frac{1}{3} x^{3}]_{- 2}^{3})$

Step 5

Simplify the answer.

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Step 5.1

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

$36 x]_{- 2}^{3} - (\frac{x^{3}}{3}]_{- 2}^{3})$

Step 5.2

Substitute and simplify.

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Step 5.2.1

Evaluate $36 x$ at $3$ and at $- 2$ .

$(36 \cdot 3) - 36 \cdot - 2 - (\frac{x^{3}}{3}]_{- 2}^{3})$

Step 5.2.2

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

$36 \cdot 3 - 36 \cdot - 2 - (\frac{3^{3}}{3} - \frac{{(- 2)}^{3}}{3})$

Step 5.2.3

Simplify.

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Step 5.2.3.1

Multiply $36$ by $3$ .

$108 - 36 \cdot - 2 - (\frac{3^{3}}{3} - \frac{{(- 2)}^{3}}{3})$

Step 5.2.3.2

Multiply $- 36$ by $- 2$ .

$108 + 72 - (\frac{3^{3}}{3} - \frac{{(- 2)}^{3}}{3})$

Step 5.2.3.3

Add $108$ and $72$ .

$180 - (\frac{3^{3}}{3} - \frac{{(- 2)}^{3}}{3})$

Step 5.2.3.4

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

$180 - (\frac{27}{3} - \frac{{(- 2)}^{3}}{3})$

Step 5.2.3.5

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

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Step 5.2.3.5.1

Factor $3$ out of $27$ .

$180 - (\frac{3 \cdot 9}{3} - \frac{{(- 2)}^{3}}{3})$

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 $3$ .

$180 - (\frac{3 \cdot 9}{3 (1)} - \frac{{(- 2)}^{3}}{3})$

Step 5.2.3.5.2.2

Cancel the common factor.

$180 - (\frac{3 \cdot 9}{3 \cdot 1} - \frac{{(- 2)}^{3}}{3})$

Step 5.2.3.5.2.3

Rewrite the expression.

$180 - (\frac{9}{1} - \frac{{(- 2)}^{3}}{3})$

Step 5.2.3.5.2.4

Divide $9$ by $1$ .

$180 - (9 - \frac{{(- 2)}^{3}}{3})$

Step 5.2.3.6

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

$180 - (9 - \frac{- 8}{3})$

Step 5.2.3.7

Move the negative in front of the fraction.

$180 - (9 - - \frac{8}{3})$

Step 5.2.3.8

Multiply $- 1$ by $- 1$ .

$180 - (9 + 1 (\frac{8}{3}))$

Step 5.2.3.9

Multiply $\frac{8}{3}$ by $1$ .

$180 - (9 + \frac{8}{3})$

Step 5.2.3.10

To write $9$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$180 - (9 \cdot \frac{3}{3} + \frac{8}{3})$

Step 5.2.3.11

Combine $9$ and $\frac{3}{3}$ .

$180 - (\frac{9 \cdot 3}{3} + \frac{8}{3})$

Step 5.2.3.12

Combine the numerators over the common denominator.

$180 - \frac{9 \cdot 3 + 8}{3}$

Step 5.2.3.13

Simplify the numerator.

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Step 5.2.3.13.1

Multiply $9$ by $3$ .

$180 - \frac{27 + 8}{3}$

Step 5.2.3.13.2

Add $27$ and $8$ .

$180 - \frac{35}{3}$

Step 5.2.3.14

To write $180$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$180 \cdot \frac{3}{3} - \frac{35}{3}$

Step 5.2.3.15

Combine $180$ and $\frac{3}{3}$ .

$\frac{180 \cdot 3}{3} - \frac{35}{3}$

Step 5.2.3.16

Combine the numerators over the common denominator.

$\frac{180 \cdot 3 - 35}{3}$

Step 5.2.3.17

Simplify the numerator.

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Step 5.2.3.17.1

Multiply $180$ by $3$ .

$\frac{540 - 35}{3}$

Step 5.2.3.17.2

Subtract $35$ from $540$ .

$\frac{505}{3}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$\frac{505}{3}$

Decimal Form:

$168.\overline{3}$

Mixed Number Form:

$168 \frac{1}{3}$

Step 7