Calculus Examples

$\int_{0}^{4} {(\sqrt{x})}^{2} d x$

Step 1

Rewrite ${(\sqrt{x})}^{2}$ as $x$ .

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Step 1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x}$ as $x^{\frac{1}{2}}$ .

$\int_{0}^{4} {(x^{\frac{1}{2}})}^{2} d x$

Step 1.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\int_{0}^{4} x^{\frac{1}{2} \cdot 2} d x$

Step 1.3

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

$\int_{0}^{4} x^{\frac{2}{2}} d x$

Step 1.4

Cancel the common factor of $2$ .

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Step 1.4.1

Cancel the common factor.

$\int_{0}^{4} x^{\frac{2}{2}} d x$

Step 1.4.2

Rewrite the expression.

$\int_{0}^{4} x^{1} d x$

Step 1.5

Simplify.

$\int_{0}^{4} x d x$

Step 2

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

$\frac{1}{2} x^{2}]_{0}^{4}$

Step 3

Substitute and simplify.

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Step 3.1

Evaluate $\frac{1}{2} x^{2}$ at $4$ and at $0$ .

$(\frac{1}{2} \cdot 4^{2}) - \frac{1}{2} \cdot 0^{2}$

Step 3.2

Simplify.

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Step 3.2.1

Raise $4$ to the power of $2$ .

$\frac{1}{2} \cdot 16 - \frac{1}{2} \cdot 0^{2}$

Step 3.2.2

Combine $\frac{1}{2}$ and $16$ .

$\frac{16}{2} - \frac{1}{2} \cdot 0^{2}$

Step 3.2.3

Cancel the common factor of $16$ and $2$ .

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Step 3.2.3.1

Factor $2$ out of $16$ .

$\frac{2 \cdot 8}{2} - \frac{1}{2} \cdot 0^{2}$

Step 3.2.3.2

Cancel the common factors.

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Step 3.2.3.2.1

Factor $2$ out of $2$ .

$\frac{2 \cdot 8}{2 (1)} - \frac{1}{2} \cdot 0^{2}$

Step 3.2.3.2.2

Cancel the common factor.

$\frac{2 \cdot 8}{2 \cdot 1} - \frac{1}{2} \cdot 0^{2}$

Step 3.2.3.2.3

Rewrite the expression.

$\frac{8}{1} - \frac{1}{2} \cdot 0^{2}$

Step 3.2.3.2.4

Divide $8$ by $1$ .

$8 - \frac{1}{2} \cdot 0^{2}$

Step 3.2.4

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

$8 - \frac{1}{2} \cdot 0$

Step 3.2.5

Multiply $0$ by $- 1$ .

$8 + 0 (\frac{1}{2})$

Step 3.2.6

Multiply $0$ by $\frac{1}{2}$ .

$8 + 0$

Step 3.2.7

Add $8$ and $0$ .

$8$

Step 4