Calculus Examples

$\int_{3}^{6} x^{2} d x$

Step 1

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

$\frac{1}{3} x^{3}]_{3}^{6}$

Step 2

Substitute and simplify.

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Step 2.1

Evaluate $\frac{1}{3} x^{3}$ at $6$ and at $3$ .

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

Step 2.2

Simplify.

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Step 2.2.1

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

$\frac{1}{3} \cdot 216 - \frac{1}{3} \cdot 3^{3}$

Step 2.2.2

Combine $\frac{1}{3}$ and $216$ .

$\frac{216}{3} - \frac{1}{3} \cdot 3^{3}$

Step 2.2.3

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

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Step 2.2.3.1

Factor $3$ out of $216$ .

$\frac{3 \cdot 72}{3} - \frac{1}{3} \cdot 3^{3}$

Step 2.2.3.2

Cancel the common factors.

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Step 2.2.3.2.1

Factor $3$ out of $3$ .

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

Step 2.2.3.2.2

Cancel the common factor.

$\frac{3 \cdot 72}{3 \cdot 1} - \frac{1}{3} \cdot 3^{3}$

Step 2.2.3.2.3

Rewrite the expression.

$\frac{72}{1} - \frac{1}{3} \cdot 3^{3}$

Step 2.2.3.2.4

Divide $72$ by $1$ .

$72 - \frac{1}{3} \cdot 3^{3}$

Step 2.2.4

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

$72 - \frac{1}{3} \cdot 27$

Step 2.2.5

Multiply $27$ by $- 1$ .

$72 - 27 (\frac{1}{3})$

Step 2.2.6

Combine $- 27$ and $\frac{1}{3}$ .

$72 + \frac{- 27}{3}$

Step 2.2.7

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

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Step 2.2.7.1

Factor $3$ out of $- 27$ .

$72 + \frac{3 \cdot - 9}{3}$

Step 2.2.7.2

Cancel the common factors.

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Step 2.2.7.2.1

Factor $3$ out of $3$ .

$72 + \frac{3 \cdot - 9}{3 (1)}$

Step 2.2.7.2.2

Cancel the common factor.

$72 + \frac{3 \cdot - 9}{3 \cdot 1}$

Step 2.2.7.2.3

Rewrite the expression.

$72 + \frac{- 9}{1}$

Step 2.2.7.2.4

Divide $- 9$ by $1$ .

$72 - 9$

Step 2.2.8

Subtract $9$ from $72$ .

$63$

Step 3

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