Calculus Examples

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

Step 1

Split the single integral into multiple integrals.

$\int_{0}^{2} 4 d x + \int_{0}^{2} - x^{2} d x$

Step 2

Apply the constant rule.

$4 x]_{0}^{2} + \int_{0}^{2} - x^{2} d x$

Step 3

Since $- 1$ is constant with respect to $x$ , move $- 1$ out of the integral.

$4 x]_{0}^{2} - \int_{0}^{2} x^{2} d x$

Step 4

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

$4 x]_{0}^{2} - (\frac{1}{3} x^{3}]_{0}^{2})$

Step 5

Simplify the answer.

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Step 5.1

Combine $\frac{1}{3}$ and $x^{3}$ .

$4 x]_{0}^{2} - (\frac{x^{3}}{3}]_{0}^{2})$

Step 5.2

Substitute and simplify.

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Step 5.2.1

Evaluate $4 x$ at $2$ and at $0$ .

$(4 \cdot 2) - 4 \cdot 0 - (\frac{x^{3}}{3}]_{0}^{2})$

Step 5.2.2

Evaluate $\frac{x^{3}}{3}$ at $2$ and at $0$ .

$4 \cdot 2 - 4 \cdot 0 - (\frac{2^{3}}{3} - \frac{0^{3}}{3})$

Step 5.2.3

Simplify.

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Step 5.2.3.1

Multiply $4$ by $2$ .

$8 - 4 \cdot 0 - (\frac{2^{3}}{3} - \frac{0^{3}}{3})$

Step 5.2.3.2

Multiply $- 4$ by $0$ .

$8 + 0 - (\frac{2^{3}}{3} - \frac{0^{3}}{3})$

Step 5.2.3.3

Add $8$ and $0$ .

$8 - (\frac{2^{3}}{3} - \frac{0^{3}}{3})$

Step 5.2.3.4

Raise $2$ to the power of $3$ .

$8 - (\frac{8}{3} - \frac{0^{3}}{3})$

Step 5.2.3.5

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

$8 - (\frac{8}{3} - \frac{0}{3})$

Step 5.2.3.6

Cancel the common factor of $0$ and $3$ .

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Step 5.2.3.6.1

Factor $3$ out of $0$ .

$8 - (\frac{8}{3} - \frac{3 (0)}{3})$

Step 5.2.3.6.2

Cancel the common factors.

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Step 5.2.3.6.2.1

Factor $3$ out of $3$ .

$8 - (\frac{8}{3} - \frac{3 \cdot 0}{3 \cdot 1})$

Step 5.2.3.6.2.2

Cancel the common factor.

$8 - (\frac{8}{3} - \frac{3 \cdot 0}{3 \cdot 1})$

Step 5.2.3.6.2.3

Rewrite the expression.

$8 - (\frac{8}{3} - \frac{0}{1})$

Step 5.2.3.6.2.4

Divide $0$ by $1$ .

$8 - (\frac{8}{3} - 0)$

Step 5.2.3.7

Multiply $- 1$ by $0$ .

$8 - (\frac{8}{3} + 0)$

Step 5.2.3.8

Add $\frac{8}{3}$ and $0$ .

$8 - \frac{8}{3}$

Step 5.2.3.9

To write $8$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$8 \cdot \frac{3}{3} - \frac{8}{3}$

Step 5.2.3.10

Combine $8$ and $\frac{3}{3}$ .

$\frac{8 \cdot 3}{3} - \frac{8}{3}$

Step 5.2.3.11

Combine the numerators over the common denominator.

$\frac{8 \cdot 3 - 8}{3}$

Step 5.2.3.12

Simplify the numerator.

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Step 5.2.3.12.1

Multiply $8$ by $3$ .

$\frac{24 - 8}{3}$

Step 5.2.3.12.2

Subtract $8$ from $24$ .

$\frac{16}{3}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$\frac{16}{3}$

Decimal Form:

$5.\overline{3}$

Mixed Number Form:

$5 \frac{1}{3}$

Step 7