Calculus Examples

$16 p \int_{0}^{5} {(1 - \frac{x}{5})}^{2} d x$

Step 1

Let $u = 1 - \frac{x}{5}$ . Then $d u = - \frac{1}{5} d x$ , so $- 5 d u = d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 1.1

Let $u = 1 - \frac{x}{5}$ . Find $\frac{d u}{d x}$ .

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Step 1.1.1

Differentiate $1 - \frac{x}{5}$ .

$\frac{d}{d x} [1 - \frac{x}{5}]$

Step 1.1.2

Differentiate.

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Step 1.1.2.1

By the Sum Rule, the derivative of $1 - \frac{x}{5}$ with respect to $x$ is $\frac{d}{d x} [1] + \frac{d}{d x} [- \frac{x}{5}]$ .

$\frac{d}{d x} [1] + \frac{d}{d x} [- \frac{x}{5}]$

Step 1.1.2.2

Since $1$ is constant with respect to $x$ , the derivative of $1$ with respect to $x$ is $0$ .

$0 + \frac{d}{d x} [- \frac{x}{5}]$

Step 1.1.3

Evaluate $\frac{d}{d x} [- \frac{x}{5}]$ .

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Step 1.1.3.1

Since $- \frac{1}{5}$ is constant with respect to $x$ , the derivative of $- \frac{x}{5}$ with respect to $x$ is $- \frac{1}{5} \frac{d}{d x} [x]$ .

$0 - \frac{1}{5} \frac{d}{d x} [x]$

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 $n = 1$ .

$0 - \frac{1}{5} \cdot 1$

Step 1.1.3.3

Multiply $- 1$ by $1$ .

$0 - \frac{1}{5}$

Step 1.1.4

Subtract $\frac{1}{5}$ from $0$ .

$- \frac{1}{5}$

Step 1.2

Substitute the lower limit in for $x$ in $u = 1 - \frac{x}{5}$ .

$u_{lower} = 1 - \frac{0}{5}$

Step 1.3

Simplify.

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Step 1.3.1

Simplify each term.

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Step 1.3.1.1

Divide $0$ by $5$ .

$u_{lower} = 1 - 0$

Step 1.3.1.2

Multiply $- 1$ by $0$ .

$u_{lower} = 1 + 0$

Step 1.3.2

Add $1$ and $0$ .

$u_{lower} = 1$

Step 1.4

Substitute the upper limit in for $x$ in $u = 1 - \frac{x}{5}$ .

$u_{upper} = 1 - \frac{5}{5}$

Step 1.5

Simplify.

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Step 1.5.1

Simplify each term.

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Step 1.5.1.1

Cancel the common factor of $5$ .

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Step 1.5.1.1.1

Cancel the common factor.

$u_{upper} = 1 - \frac{5}{5}$

Step 1.5.1.1.2

Rewrite the expression.

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

Step 1.5.1.2

Multiply $- 1$ by $1$ .

$u_{upper} = 1 - 1$

Step 1.5.2

Subtract $1$ from $1$ .

$u_{upper} = 0$

Step 1.6

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

$u_{lower} = 1$

$u_{upper} = 0$

Step 1.7

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

$16 p \int_{1}^{0} - u^{2} \frac{- 1}{- \frac{1}{5}} d u$

Step 2

Simplify.

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Step 2.1

Dividing two negative values results in a positive value.

$16 p \int_{1}^{0} - u^{2} \frac{1}{\frac{1}{5}} d u$

Step 2.2

Multiply by the reciprocal of the fraction to divide by $\frac{1}{5}$ .

$16 p \int_{1}^{0} - u^{2} (1 \cdot 5) d u$

Step 2.3

Multiply $5$ by $1$ .

$16 p \int_{1}^{0} - u^{2} \cdot 5 d u$

Step 2.4

Multiply $5$ by $- 1$ .

$16 p \int_{1}^{0} - 5 u^{2} d u$

Step 3

Since $- 5$ is constant with respect to $u$ , move $- 5$ out of the integral.

$16 p (- 5 \int_{1}^{0} u^{2} d u)$

Step 4

Multiply $- 5$ by $16$ .

$- 80 p \int_{1}^{0} u^{2} d u$

Step 5

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

$- 80 p \frac{1}{3} u^{3}]_{1}^{0}$

Step 6

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

$- 80 p \frac{u^{3}}{3}]_{1}^{0}$

Step 7

Substitute and simplify.

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Step 7.1

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

$- 80 p ((\frac{0^{3}}{3}) - \frac{1^{3}}{3})$

Step 7.2

Simplify.

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Step 7.2.1

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

$- 80 p (\frac{0}{3} - \frac{1^{3}}{3})$

Step 7.2.2

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

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Step 7.2.2.1

Factor $3$ out of $0$ .

$- 80 p (\frac{3 (0)}{3} - \frac{1^{3}}{3})$

Step 7.2.2.2

Cancel the common factors.

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Step 7.2.2.2.1

Factor $3$ out of $3$ .

$- 80 p (\frac{3 \cdot 0}{3 \cdot 1} - \frac{1^{3}}{3})$

Step 7.2.2.2.2

Cancel the common factor.

$- 80 p (\frac{3 \cdot 0}{3 \cdot 1} - \frac{1^{3}}{3})$

Step 7.2.2.2.3

Rewrite the expression.

$- 80 p (\frac{0}{1} - \frac{1^{3}}{3})$

Step 7.2.2.2.4

Divide $0$ by $1$ .

$- 80 p (0 - \frac{1^{3}}{3})$

Step 7.2.3

One to any power is one.

$- 80 p (0 - \frac{1}{3})$

Step 7.2.4

Subtract $\frac{1}{3}$ from $0$ .

$- 80 p (- \frac{1}{3})$

Step 7.2.5

Multiply $- 1$ by $- 80$ .

$80 p \frac{1}{3}$

Step 7.2.6

Combine $80$ and $\frac{1}{3}$ .

$p \frac{80}{3}$

Step 7.2.7

Combine $p$ and $\frac{80}{3}$ .

$\frac{p \cdot 80}{3}$

Step 7.2.8

Move $80$ to the left of $p$ .

$\frac{80 p}{3}$

Step 8

Reorder terms.

$\frac{80}{3} p$

Step 9

Combine $\frac{80}{3}$ and $p$ .

$\frac{80 p}{3}$

Step 10