Calculus Examples

$\int_{1}^{8} r^{2} d r$

Step 1

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

$\frac{1}{3} r^{3}]_{1}^{8}$

Step 2

Substitute and simplify.

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Step 2.1

Evaluate $\frac{1}{3} r^{3}$ at $8$ and at $1$ .

$(\frac{1}{3} \cdot 8^{3}) - \frac{1}{3} \cdot 1^{3}$

Step 2.2

Simplify.

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Step 2.2.1

Raise $8$ to the power of $3$ .

$\frac{1}{3} \cdot 512 - \frac{1}{3} \cdot 1^{3}$

Step 2.2.2

Combine $\frac{1}{3}$ and $512$ .

$\frac{512}{3} - \frac{1}{3} \cdot 1^{3}$

Step 2.2.3

One to any power is one.

$\frac{512}{3} - \frac{1}{3} \cdot 1$

Step 2.2.4

Multiply $- 1$ by $1$ .

$\frac{512}{3} - \frac{1}{3}$

Step 2.2.5

Combine the numerators over the common denominator.

$\frac{512 - 1}{3}$

Step 2.2.6

Subtract $1$ from $512$ .

$\frac{511}{3}$

Step 3

The result can be shown in multiple forms.

Exact Form:

$\frac{511}{3}$

Decimal Form:

$170.\overline{3}$

Mixed Number Form:

$170 \frac{1}{3}$

Step 4