Calculus Examples

$π \int_{0}^{6} {(- x + 6)}^{2} d x$

Step 1

Let $u = - x + 6$ . Then $d u = - d x$ , so $- d u = d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 1.1

Let $u = - x + 6$ . Find $\frac{d u}{d x}$ .

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Step 1.1.1

Differentiate $- x + 6$ .

$\frac{d}{d x} [- x + 6]$

Step 1.1.2

By the Sum Rule, the derivative of $- x + 6$ with respect to $x$ is $\frac{d}{d x} [- x] + \frac{d}{d x} [6]$ .

$\frac{d}{d x} [- x] + \frac{d}{d x} [6]$

Step 1.1.3

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

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Step 1.1.3.1

Since $- 1$ is constant with respect to $x$ , the derivative of $- x$ with respect to $x$ is $- \frac{d}{d x} [x]$ .

$- \frac{d}{d x} [x] + \frac{d}{d x} [6]$

Step 1.1.3.2

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

$- 1 \cdot 1 + \frac{d}{d x} [6]$

Step 1.1.3.3

Multiply $- 1$ by $1$ .

$- 1 + \frac{d}{d x} [6]$

Step 1.1.4

Differentiate using the Constant Rule.

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Step 1.1.4.1

Since $6$ is constant with respect to $x$ , the derivative of $6$ with respect to $x$ is $0$ .

$- 1 + 0$

Step 1.1.4.2

Add $- 1$ and $0$ .

$- 1$

Step 1.2

Substitute the lower limit in for $x$ in $u = - x + 6$ .

$u_{lower} = - 0 + 6$

Step 1.3

Simplify.

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Step 1.3.1

Multiply $- 1$ by $0$ .

$u_{lower} = 0 + 6$

Step 1.3.2

Add $0$ and $6$ .

$u_{lower} = 6$

Step 1.4

Substitute the upper limit in for $x$ in $u = - x + 6$ .

$u_{upper} = - 1 \cdot 6 + 6$

Step 1.5

Simplify.

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Step 1.5.1

Multiply $- 1$ by $6$ .

$u_{upper} = - 6 + 6$

Step 1.5.2

Add $- 6$ and $6$ .

$u_{upper} = 0$

Step 1.6

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

$u_{lower} = 6$

$u_{upper} = 0$

Step 1.7

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

$π \int_{6}^{0} - u^{2} d u$

Step 2

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

$π (- \int_{6}^{0} u^{2} d u)$

Step 3

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

$π (- (\frac{1}{3} u^{3}]_{6}^{0}))$

Step 4

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

$π (- (\frac{u^{3}}{3}]_{6}^{0}))$

Step 5

Substitute and simplify.

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Step 5.1

Evaluate $\frac{u^{3}}{3}$ at $0$ and at $6$ .

$π (- ((\frac{0^{3}}{3}) - \frac{6^{3}}{3}))$

Step 5.2

Simplify.

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Step 5.2.1

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

$π (- (\frac{0}{3} - \frac{6^{3}}{3}))$

Step 5.2.2

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

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Step 5.2.2.1

Factor $3$ out of $0$ .

$π (- (\frac{3 (0)}{3} - \frac{6^{3}}{3}))$

Step 5.2.2.2

Cancel the common factors.

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Step 5.2.2.2.1

Factor $3$ out of $3$ .

$π (- (\frac{3 \cdot 0}{3 \cdot 1} - \frac{6^{3}}{3}))$

Step 5.2.2.2.2

Cancel the common factor.

$π (- (\frac{3 \cdot 0}{3 \cdot 1} - \frac{6^{3}}{3}))$

Step 5.2.2.2.3

Rewrite the expression.

$π (- (\frac{0}{1} - \frac{6^{3}}{3}))$

Step 5.2.2.2.4

Divide $0$ by $1$ .

$π (- (0 - \frac{6^{3}}{3}))$

Step 5.2.3

Raise $6$ to the power of $3$ .

$π (- (0 - \frac{216}{3}))$

Step 5.2.4

Cancel the common factor of $216$ and $3$ .

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Step 5.2.4.1

Factor $3$ out of $216$ .

$π (- (0 - \frac{3 \cdot 72}{3}))$

Step 5.2.4.2

Cancel the common factors.

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Step 5.2.4.2.1

Factor $3$ out of $3$ .

$π (- (0 - \frac{3 \cdot 72}{3 (1)}))$

Step 5.2.4.2.2

Cancel the common factor.

$π (- (0 - \frac{3 \cdot 72}{3 \cdot 1}))$

Step 5.2.4.2.3

Rewrite the expression.

$π (- (0 - \frac{72}{1}))$

Step 5.2.4.2.4

Divide $72$ by $1$ .

$π (- (0 - 1 \cdot 72))$

Step 5.2.5

Multiply $- 1$ by $72$ .

$π (- (0 - 72))$

Step 5.2.6

Subtract $72$ from $0$ .

$π (- - 72)$

Step 5.2.7

Multiply $- 1$ by $- 72$ .

$π \cdot 72$

Step 5.2.8

Move $72$ to the left of $π$ .

$72 π$

Step 6

The result can be shown in multiple forms.

Exact Form:

$72 π$

Decimal Form:

$226.19467105 \dots$

Step 7