Calculus Examples

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

Step 1

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 2

Substitute and simplify.

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Step 2.1

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

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

Step 2.2

Simplify.

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Step 2.2.1

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

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

Step 2.2.2

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

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

Step 2.2.3

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

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Step 2.2.3.1

Factor $2$ out of $16$ .

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

Step 2.2.3.2

Cancel the common factors.

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Step 2.2.3.2.1

Factor $2$ out of $2$ .

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

Step 2.2.3.2.2

Cancel the common factor.

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

Step 2.2.3.2.3

Rewrite the expression.

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

Step 2.2.3.2.4

Divide $8$ by $1$ .

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

Step 2.2.4

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

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

Step 2.2.5

Multiply $0$ by $- 1$ .

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

Step 2.2.6

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

$8 + 0$

Step 2.2.7

Add $8$ and $0$ .

$8$

Step 3