Calculus Examples

$\int_{0}^{2} π {(4 - 2 x)}^{2} d x$

Step 1

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

$π \int_{0}^{2} {(4 - 2 x)}^{2} d x$

Step 2

Let $u = 4 - 2 x$ . Then $d u = - 2 d x$ , so $- \frac{1}{2} d u = d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 2.1

Let $u = 4 - 2 x$ . Find $\frac{d u}{d x}$ .

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Step 2.1.1

Differentiate $4 - 2 x$ .

$\frac{d}{d x} [4 - 2 x]$

Step 2.1.2

Differentiate.

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Step 2.1.2.1

By the Sum Rule, the derivative of $4 - 2 x$ with respect to $x$ is $\frac{d}{d x} [4] + \frac{d}{d x} [- 2 x]$ .

$\frac{d}{d x} [4] + \frac{d}{d x} [- 2 x]$

Step 2.1.2.2

Since $4$ is constant with respect to $x$ , the derivative of $4$ with respect to $x$ is $0$ .

$0 + \frac{d}{d x} [- 2 x]$

Step 2.1.3

Evaluate $\frac{d}{d x} [- 2 x]$ .

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Step 2.1.3.1

Since $- 2$ is constant with respect to $x$ , the derivative of $- 2 x$ with respect to $x$ is $- 2 \frac{d}{d x} [x]$ .

$0 - 2 \frac{d}{d x} [x]$

Step 2.1.3.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$0 - 2 \cdot 1$

Step 2.1.3.3

Multiply $- 2$ by $1$ .

$0 - 2$

Step 2.1.4

Subtract $2$ from $0$ .

$- 2$

Step 2.2

Substitute the lower limit in for $x$ in $u = 4 - 2 x$ .

$u_{lower} = 4 - 2 \cdot 0$

Step 2.3

Simplify.

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Step 2.3.1

Multiply $- 2$ by $0$ .

$u_{lower} = 4 + 0$

Step 2.3.2

Add $4$ and $0$ .

$u_{lower} = 4$

Step 2.4

Substitute the upper limit in for $x$ in $u = 4 - 2 x$ .

$u_{upper} = 4 - 2 \cdot 2$

Step 2.5

Simplify.

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Step 2.5.1

Multiply $- 2$ by $2$ .

$u_{upper} = 4 - 4$

Step 2.5.2

Subtract $4$ from $4$ .

$u_{upper} = 0$

Step 2.6

The values found for $u_{lower}$ and $u_{upper}$ will be used to evaluate the definite integral.

$u_{lower} = 4$

$u_{upper} = 0$

Step 2.7

Rewrite the problem using $u$ , $d u$ , and the new limits of integration.

$π \int_{4}^{0} u^{2} \frac{1}{- 2} d u$

Step 3

Simplify.

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Step 3.1

Move the negative in front of the fraction.

$π \int_{4}^{0} u^{2} (- \frac{1}{2}) d u$

Step 3.2

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

$π \int_{4}^{0} - \frac{u^{2}}{2} d u$

Step 4

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

$π (- \int_{4}^{0} \frac{u^{2}}{2} d u)$

Step 5

Since $\frac{1}{2}$ is constant with respect to $u$ , move $\frac{1}{2}$ out of the integral.

$π (- (\frac{1}{2} \int_{4}^{0} u^{2} d u))$

Step 6

Combine $π$ and $\frac{1}{2}$ .

$- \frac{π}{2} \int_{4}^{0} u^{2} d u$

Step 7

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

$- \frac{π}{2} \frac{1}{3} u^{3}]_{4}^{0}$

Step 8

Substitute and simplify.

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Step 8.1

Evaluate $\frac{1}{3} u^{3}$ at $0$ and at $4$ .

$- \frac{π}{2} ((\frac{1}{3} \cdot 0^{3}) - \frac{1}{3} \cdot 4^{3})$

Step 8.2

Simplify.

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Step 8.2.1

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

$- \frac{π}{2} (\frac{1}{3} \cdot 0 - \frac{1}{3} \cdot 4^{3})$

Step 8.2.2

Multiply $\frac{1}{3}$ by $0$ .

$- \frac{π}{2} (0 - \frac{1}{3} \cdot 4^{3})$

Step 8.2.3

Raise $4$ to the power of $3$ .

$- \frac{π}{2} (0 - \frac{1}{3} \cdot 64)$

Step 8.2.4

Multiply $64$ by $- 1$ .

$- \frac{π}{2} (0 - 64 (\frac{1}{3}))$

Step 8.2.5

Combine $- 64$ and $\frac{1}{3}$ .

$- \frac{π}{2} (0 + \frac{- 64}{3})$

Step 8.2.6

Move the negative in front of the fraction.

$- \frac{π}{2} (0 - \frac{64}{3})$

Step 8.2.7

Subtract $\frac{64}{3}$ from $0$ .

$- \frac{π}{2} (- \frac{64}{3})$

Step 8.2.8

Multiply $- 1$ by $- 1$ .

$1 \frac{π}{2} \frac{64}{3}$

Step 8.2.9

Multiply $\frac{π}{2}$ by $1$ .

$\frac{π}{2} \cdot \frac{64}{3}$

Step 8.2.10

Multiply $\frac{π}{2}$ by $\frac{64}{3}$ .

$\frac{π \cdot 64}{2 \cdot 3}$

Step 8.2.11

Multiply $2$ by $3$ .

$\frac{π \cdot 64}{6}$

Step 8.2.12

Move $64$ to the left of $π$ .

$\frac{64 \cdot π}{6}$

Step 8.2.13

Cancel the common factor of $64$ and $6$ .

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Step 8.2.13.1

Factor $2$ out of $64 π$ .

$\frac{2 (32 π)}{6}$

Step 8.2.13.2

Cancel the common factors.

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Step 8.2.13.2.1

Factor $2$ out of $6$ .

$\frac{2 (32 π)}{2 (3)}$

Step 8.2.13.2.2

Cancel the common factor.

$\frac{2 (32 π)}{2 \cdot 3}$

Step 8.2.13.2.3

Rewrite the expression.

$\frac{32 π}{3}$

Step 9

The result can be shown in multiple forms.

Exact Form:

$\frac{32 π}{3}$

Decimal Form:

$33.51032163 \dots$

Step 10