Calculus Examples

$\int_{0}^{4 \sqrt{3}} p (\frac{y^{4}}{256}) d y$

Step 1

Combine $p$ and $\frac{y^{4}}{256}$ .

$\int_{0}^{4 \sqrt{3}} \frac{p y^{4}}{256} d y$

Step 2

Since $\frac{p}{256}$ is constant with respect to $y$ , move $\frac{p}{256}$ out of the integral.

$\frac{p}{256} \int_{0}^{4 \sqrt{3}} y^{4} d y$

Step 3

By the Power Rule, the integral of $y^{4}$ with respect to $y$ is $\frac{1}{5} y^{5}$ .

$\frac{p}{256} \frac{1}{5} y^{5}]_{0}^{4 \sqrt{3}}$

Step 4

Simplify the answer.

Tap for more steps...

Step 4.1

Simplify.

Tap for more steps...

Step 4.1.1

Combine $\frac{1}{5}$ and $y^{5}$ .

$\frac{p}{256} \frac{y^{5}}{5}]_{0}^{4 \sqrt{3}}$

Step 4.1.2

Combine $\frac{p}{256}$ and $\frac{y^{5}}{5}]_{0}^{4 \sqrt{3}}$ .

$\frac{p \frac{y^{5}}{5}]_{0}^{4 \sqrt{3}}}{256}$

Step 4.2

Substitute and simplify.

Tap for more steps...

Step 4.2.1

Evaluate $\frac{y^{5}}{5}$ at $4 \sqrt{3}$ and at $0$ .

$\frac{p ((\frac{{(4 \sqrt{3})}^{5}}{5}) - \frac{0^{5}}{5})}{256}$

Step 4.2.2

Simplify.

Tap for more steps...

Step 4.2.2.1

Factor $4$ out of $4 \sqrt{3}$ .

$\frac{p (\frac{{(4 (\sqrt{3}))}^{5}}{5} - \frac{0^{5}}{5})}{256}$

Step 4.2.2.2

Apply the product rule to $4 (\sqrt{3})$ .

$\frac{p (\frac{4^{5} {\sqrt{3}}^{5}}{5} - \frac{0^{5}}{5})}{256}$

Step 4.2.2.3

Raise $4$ to the power of $5$ .

$\frac{p (\frac{1024 {\sqrt{3}}^{5}}{5} - \frac{0^{5}}{5})}{256}$

Step 4.2.2.4

Rewrite ${\sqrt{3}}^{5}$ as $\sqrt{3^{5}}$ .

$\frac{p (\frac{1024 \sqrt{3^{5}}}{5} - \frac{0^{5}}{5})}{256}$

Step 4.2.2.5

Raise $3$ to the power of $5$ .

$\frac{p (\frac{1024 \sqrt{243}}{5} - \frac{0^{5}}{5})}{256}$

Step 4.2.2.6

Raising $0$ to any positive power yields $0$ .

$\frac{p (\frac{1024 \sqrt{243}}{5} - \frac{0}{5})}{256}$

Step 4.2.2.7

Cancel the common factor of $0$ and $5$ .

Tap for more steps...

Step 4.2.2.7.1

Factor $5$ out of $0$ .

$\frac{p (\frac{1024 \sqrt{243}}{5} - \frac{5 (0)}{5})}{256}$

Step 4.2.2.7.2

Cancel the common factors.

Tap for more steps...

Step 4.2.2.7.2.1

Factor $5$ out of $5$ .

$\frac{p (\frac{1024 \sqrt{243}}{5} - \frac{5 \cdot 0}{5 \cdot 1})}{256}$

Step 4.2.2.7.2.2

Cancel the common factor.

$\frac{p (\frac{1024 \sqrt{243}}{5} - \frac{5 \cdot 0}{5 \cdot 1})}{256}$

Step 4.2.2.7.2.3

Rewrite the expression.

$\frac{p (\frac{1024 \sqrt{243}}{5} - \frac{0}{1})}{256}$

Step 4.2.2.7.2.4

Divide $0$ by $1$ .

$\frac{p (\frac{1024 \sqrt{243}}{5} - 0)}{256}$

Step 4.2.2.8

Multiply $- 1$ by $0$ .

$\frac{p (\frac{1024 \sqrt{243}}{5} + 0)}{256}$

Step 4.2.2.9

Add $\frac{1024 \sqrt{243}}{5}$ and $0$ .

$\frac{p \frac{1024 \sqrt{243}}{5}}{256}$

Step 4.2.2.10

Combine $p$ and $\frac{1024 \sqrt{243}}{5}$ .

$\frac{\frac{p (1024 \sqrt{243})}{5}}{256}$

Step 4.2.2.11

Move $1024$ to the left of $p$ .

$\frac{\frac{1024 \cdot p \sqrt{243}}{5}}{256}$

Step 4.2.2.12

Rewrite $\frac{\frac{1024 p \sqrt{243}}{5}}{256}$ as a product.

$\frac{1024 p \sqrt{243}}{5} \cdot \frac{1}{256}$

Step 4.2.2.13

Multiply $\frac{1024 p \sqrt{243}}{5}$ by $\frac{1}{256}$ .

$\frac{1024 p \sqrt{243}}{5 \cdot 256}$

Step 4.2.2.14

Multiply $5$ by $256$ .

$\frac{1024 p \sqrt{243}}{1280}$

Step 4.2.2.15

Cancel the common factor of $1024$ and $1280$ .

Tap for more steps...

Step 4.2.2.15.1

Factor $256$ out of $1024 p \sqrt{243}$ .

$\frac{256 (4 p \sqrt{243})}{1280}$

Step 4.2.2.15.2

Cancel the common factors.

Tap for more steps...

Step 4.2.2.15.2.1

Factor $256$ out of $1280$ .

$\frac{256 (4 p \sqrt{243})}{256 (5)}$

Step 4.2.2.15.2.2

Cancel the common factor.

$\frac{256 (4 p \sqrt{243})}{256 \cdot 5}$

Step 4.2.2.15.2.3

Rewrite the expression.

$\frac{4 p \sqrt{243}}{5}$

Step 4.3

Simplify.

Tap for more steps...

Step 4.3.1

Rewrite $243$ as $9^{2} \cdot 3$ .

Tap for more steps...

Step 4.3.1.1

Factor $81$ out of $243$ .

$\frac{4 p \sqrt{81 (3)}}{5}$

Step 4.3.1.2

Rewrite $81$ as $9^{2}$ .

$\frac{4 p \sqrt{9^{2} \cdot 3}}{5}$

Step 4.3.2

Pull terms out from under the radical.

$\frac{4 p (9 \sqrt{3})}{5}$

Step 4.3.3

Multiply $9$ by $4$ .

$\frac{36 p \sqrt{3}}{5}$

Step 4.4

Reorder terms.

$\frac{36 \sqrt{3}}{5} p$

Step 5

Combine $\frac{36 \sqrt{3}}{5}$ and $p$ .

$\frac{36 \sqrt{3} p}{5}$

Step 6