Calculus Examples

$\frac{1}{3} \cdot (p r h^{2})$

Step 1

Simplify.

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Step 1.1

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

$\int \frac{p}{3} \cdot (r h^{2}) d p$

Step 1.2

Combine $r$ and $\frac{p}{3}$ .

$\int \frac{r p}{3} \cdot h^{2} d p$

Step 1.3

Combine $\frac{r p}{3}$ and $h^{2}$ .

$\int \frac{r p h^{2}}{3} d p$

Step 2

Since $\frac{r h^{2}}{3}$ is constant with respect to $p$ , move $\frac{r h^{2}}{3}$ out of the integral.

$\frac{r h^{2}}{3} \int p d p$

Step 3

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

$\frac{r h^{2}}{3} (\frac{1}{2} p^{2} + C)$

Step 4

Simplify the answer.

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Step 4.1

Combine $\frac{1}{2}$ and $p^{2}$ .

$\frac{r h^{2}}{3} (\frac{p^{2}}{2} + C)$

Step 4.2

Rewrite $\frac{r h^{2}}{3} (\frac{p^{2}}{2} + C)$ as $\frac{1}{3} r h^{2} \frac{1}{2} p^{2} + C$ .

$\frac{1}{3} r h^{2} \frac{1}{2} p^{2} + C$

Step 4.3

Simplify.

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Step 4.3.1

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

$\frac{r}{3} h^{2} \frac{1}{2} p^{2} + C$

Step 4.3.2

Combine $\frac{r}{3}$ and $h^{2}$ .

$\frac{r h^{2}}{3} \cdot \frac{1}{2} p^{2} + C$

Step 4.3.3

Multiply $\frac{r h^{2}}{3}$ by $\frac{1}{2}$ .

$\frac{r h^{2}}{3 \cdot 2} p^{2} + C$

Step 4.3.4

Multiply $3$ by $2$ .

$\frac{r h^{2}}{6} p^{2} + C$

Step 4.3.5

Combine $\frac{r h^{2}}{6}$ and $p^{2}$ .

$\frac{r h^{2} p^{2}}{6} + C$

$\frac{1}{6} r h^{2} p^{2} + C$