Calculus Examples

$\int_{0}^{20} | x - 10 | d x$

Step 1

Split up the integral depending on where $x - 10$ is positive and negative.

$\int_{0}^{10} - x + 10 d x + \int_{10}^{20} x - 10 d x$

Step 2

Split the single integral into multiple integrals.

$\int_{0}^{10} - x d x + \int_{0}^{10} 10 d x + \int_{10}^{20} x - 10 d x$

Step 3

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

$- \int_{0}^{10} x d x + \int_{0}^{10} 10 d x + \int_{10}^{20} x - 10 d x$

Step 4

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

$- (\frac{1}{2} x^{2}]_{0}^{10}) + \int_{0}^{10} 10 d x + \int_{10}^{20} x - 10 d x$

Step 5

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

$- (\frac{x^{2}}{2}]_{0}^{10}) + \int_{0}^{10} 10 d x + \int_{10}^{20} x - 10 d x$

Step 6

Apply the constant rule.

$- (\frac{x^{2}}{2}]_{0}^{10}) + 10 x]_{0}^{10} + \int_{10}^{20} x - 10 d x$

Step 7

Split the single integral into multiple integrals.

$- (\frac{x^{2}}{2}]_{0}^{10}) + 10 x]_{0}^{10} + \int_{10}^{20} x d x + \int_{10}^{20} - 10 d x$

Step 8

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

$- (\frac{x^{2}}{2}]_{0}^{10}) + 10 x]_{0}^{10} + \frac{1}{2} x^{2}]_{10}^{20} + \int_{10}^{20} - 10 d x$

Step 9

Apply the constant rule.

$- (\frac{x^{2}}{2}]_{0}^{10}) + 10 x]_{0}^{10} + \frac{1}{2} x^{2}]_{10}^{20} + - 10 x]_{10}^{20}$

Step 10

Simplify the answer.

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Step 10.1

Combine $\frac{1}{2} x^{2}]_{10}^{20}$ and $- 10 x]_{10}^{20}$ .

$- (\frac{x^{2}}{2}]_{0}^{10}) + 10 x]_{0}^{10} + \frac{1}{2} x^{2} - 10 x]_{10}^{20}$

Step 10.2

Substitute and simplify.

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Step 10.2.1

Evaluate $\frac{x^{2}}{2}$ at $10$ and at $0$ .

$- ((\frac{10^{2}}{2}) - \frac{0^{2}}{2}) + 10 x]_{0}^{10} + \frac{1}{2} x^{2} - 10 x]_{10}^{20}$

Step 10.2.2

Evaluate $10 x$ at $10$ and at $0$ .

$- (\frac{10^{2}}{2} - \frac{0^{2}}{2}) + 10 \cdot 10 - 10 \cdot 0 + \frac{1}{2} x^{2} - 10 x]_{10}^{20}$

Step 10.2.3

Evaluate $\frac{1}{2} x^{2} - 10 x$ at $20$ and at $10$ .

$- (\frac{10^{2}}{2} - \frac{0^{2}}{2}) + 10 \cdot 10 - 10 \cdot 0 + (\frac{1}{2} \cdot 20^{2} - 10 \cdot 20) - (\frac{1}{2} \cdot 10^{2} - 10 \cdot 10)$

Step 10.2.4

Simplify.

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Step 10.2.4.1

Raise $10$ to the power of $2$ .

$- (\frac{100}{2} - \frac{0^{2}}{2}) + 10 \cdot 10 - 10 \cdot 0 + (\frac{1}{2} \cdot 20^{2} - 10 \cdot 20) - (\frac{1}{2} \cdot 10^{2} - 10 \cdot 10)$

Step 10.2.4.2

Cancel the common factor of $100$ and $2$ .

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Step 10.2.4.2.1

Factor $2$ out of $100$ .

$- (\frac{2 \cdot 50}{2} - \frac{0^{2}}{2}) + 10 \cdot 10 - 10 \cdot 0 + (\frac{1}{2} \cdot 20^{2} - 10 \cdot 20) - (\frac{1}{2} \cdot 10^{2} - 10 \cdot 10)$

Step 10.2.4.2.2

Cancel the common factors.

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Step 10.2.4.2.2.1

Factor $2$ out of $2$ .

$- (\frac{2 \cdot 50}{2 (1)} - \frac{0^{2}}{2}) + 10 \cdot 10 - 10 \cdot 0 + (\frac{1}{2} \cdot 20^{2} - 10 \cdot 20) - (\frac{1}{2} \cdot 10^{2} - 10 \cdot 10)$

Step 10.2.4.2.2.2

Cancel the common factor.

$- (\frac{2 \cdot 50}{2 \cdot 1} - \frac{0^{2}}{2}) + 10 \cdot 10 - 10 \cdot 0 + (\frac{1}{2} \cdot 20^{2} - 10 \cdot 20) - (\frac{1}{2} \cdot 10^{2} - 10 \cdot 10)$

Step 10.2.4.2.2.3

Rewrite the expression.

$- (\frac{50}{1} - \frac{0^{2}}{2}) + 10 \cdot 10 - 10 \cdot 0 + (\frac{1}{2} \cdot 20^{2} - 10 \cdot 20) - (\frac{1}{2} \cdot 10^{2} - 10 \cdot 10)$

Step 10.2.4.2.2.4

Divide $50$ by $1$ .

$- (50 - \frac{0^{2}}{2}) + 10 \cdot 10 - 10 \cdot 0 + (\frac{1}{2} \cdot 20^{2} - 10 \cdot 20) - (\frac{1}{2} \cdot 10^{2} - 10 \cdot 10)$

Step 10.2.4.3

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

$- (50 - \frac{0}{2}) + 10 \cdot 10 - 10 \cdot 0 + (\frac{1}{2} \cdot 20^{2} - 10 \cdot 20) - (\frac{1}{2} \cdot 10^{2} - 10 \cdot 10)$

Step 10.2.4.4

Cancel the common factor of $0$ and $2$ .

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Step 10.2.4.4.1

Factor $2$ out of $0$ .

$- (50 - \frac{2 (0)}{2}) + 10 \cdot 10 - 10 \cdot 0 + (\frac{1}{2} \cdot 20^{2} - 10 \cdot 20) - (\frac{1}{2} \cdot 10^{2} - 10 \cdot 10)$

Step 10.2.4.4.2

Cancel the common factors.

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Step 10.2.4.4.2.1

Factor $2$ out of $2$ .

$- (50 - \frac{2 \cdot 0}{2 \cdot 1}) + 10 \cdot 10 - 10 \cdot 0 + (\frac{1}{2} \cdot 20^{2} - 10 \cdot 20) - (\frac{1}{2} \cdot 10^{2} - 10 \cdot 10)$

Step 10.2.4.4.2.2

Cancel the common factor.

$- (50 - \frac{2 \cdot 0}{2 \cdot 1}) + 10 \cdot 10 - 10 \cdot 0 + (\frac{1}{2} \cdot 20^{2} - 10 \cdot 20) - (\frac{1}{2} \cdot 10^{2} - 10 \cdot 10)$

Step 10.2.4.4.2.3

Rewrite the expression.

$- (50 - \frac{0}{1}) + 10 \cdot 10 - 10 \cdot 0 + (\frac{1}{2} \cdot 20^{2} - 10 \cdot 20) - (\frac{1}{2} \cdot 10^{2} - 10 \cdot 10)$

Step 10.2.4.4.2.4

Divide $0$ by $1$ .

$- (50 - 0) + 10 \cdot 10 - 10 \cdot 0 + (\frac{1}{2} \cdot 20^{2} - 10 \cdot 20) - (\frac{1}{2} \cdot 10^{2} - 10 \cdot 10)$

Step 10.2.4.5

Multiply $- 1$ by $0$ .

$- (50 + 0) + 10 \cdot 10 - 10 \cdot 0 + (\frac{1}{2} \cdot 20^{2} - 10 \cdot 20) - (\frac{1}{2} \cdot 10^{2} - 10 \cdot 10)$

Step 10.2.4.6

Add $50$ and $0$ .

$- 1 \cdot 50 + 10 \cdot 10 - 10 \cdot 0 + (\frac{1}{2} \cdot 20^{2} - 10 \cdot 20) - (\frac{1}{2} \cdot 10^{2} - 10 \cdot 10)$

Step 10.2.4.7

Multiply $- 1$ by $50$ .

$- 50 + 10 \cdot 10 - 10 \cdot 0 + (\frac{1}{2} \cdot 20^{2} - 10 \cdot 20) - (\frac{1}{2} \cdot 10^{2} - 10 \cdot 10)$

Step 10.2.4.8

Multiply $10$ by $10$ .

$- 50 + 100 - 10 \cdot 0 + (\frac{1}{2} \cdot 20^{2} - 10 \cdot 20) - (\frac{1}{2} \cdot 10^{2} - 10 \cdot 10)$

Step 10.2.4.9

Multiply $- 10$ by $0$ .

$- 50 + 100 + 0 + (\frac{1}{2} \cdot 20^{2} - 10 \cdot 20) - (\frac{1}{2} \cdot 10^{2} - 10 \cdot 10)$

Step 10.2.4.10

Add $100$ and $0$ .

$- 50 + 100 + (\frac{1}{2} \cdot 20^{2} - 10 \cdot 20) - (\frac{1}{2} \cdot 10^{2} - 10 \cdot 10)$

Step 10.2.4.11

Add $- 50$ and $100$ .

$50 + (\frac{1}{2} \cdot 20^{2} - 10 \cdot 20) - (\frac{1}{2} \cdot 10^{2} - 10 \cdot 10)$

Step 10.2.4.12

Raise $20$ to the power of $2$ .

$50 + \frac{1}{2} \cdot 400 - 10 \cdot 20 - (\frac{1}{2} \cdot 10^{2} - 10 \cdot 10)$

Step 10.2.4.13

Combine $\frac{1}{2}$ and $400$ .

$50 + \frac{400}{2} - 10 \cdot 20 - (\frac{1}{2} \cdot 10^{2} - 10 \cdot 10)$

Step 10.2.4.14

Cancel the common factor of $400$ and $2$ .

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Step 10.2.4.14.1

Factor $2$ out of $400$ .

$50 + \frac{2 \cdot 200}{2} - 10 \cdot 20 - (\frac{1}{2} \cdot 10^{2} - 10 \cdot 10)$

Step 10.2.4.14.2

Cancel the common factors.

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Step 10.2.4.14.2.1

Factor $2$ out of $2$ .

$50 + \frac{2 \cdot 200}{2 (1)} - 10 \cdot 20 - (\frac{1}{2} \cdot 10^{2} - 10 \cdot 10)$

Step 10.2.4.14.2.2

Cancel the common factor.

$50 + \frac{2 \cdot 200}{2 \cdot 1} - 10 \cdot 20 - (\frac{1}{2} \cdot 10^{2} - 10 \cdot 10)$

Step 10.2.4.14.2.3

Rewrite the expression.

$50 + \frac{200}{1} - 10 \cdot 20 - (\frac{1}{2} \cdot 10^{2} - 10 \cdot 10)$

Step 10.2.4.14.2.4

Divide $200$ by $1$ .

$50 + 200 - 10 \cdot 20 - (\frac{1}{2} \cdot 10^{2} - 10 \cdot 10)$

Step 10.2.4.15

Multiply $- 10$ by $20$ .

$50 + 200 - 200 - (\frac{1}{2} \cdot 10^{2} - 10 \cdot 10)$

Step 10.2.4.16

Subtract $200$ from $200$ .

$50 + 0 - (\frac{1}{2} \cdot 10^{2} - 10 \cdot 10)$

Step 10.2.4.17

Raise $10$ to the power of $2$ .

$50 + 0 - (\frac{1}{2} \cdot 100 - 10 \cdot 10)$

Step 10.2.4.18

Combine $\frac{1}{2}$ and $100$ .

$50 + 0 - (\frac{100}{2} - 10 \cdot 10)$

Step 10.2.4.19

Cancel the common factor of $100$ and $2$ .

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Step 10.2.4.19.1

Factor $2$ out of $100$ .

$50 + 0 - (\frac{2 \cdot 50}{2} - 10 \cdot 10)$

Step 10.2.4.19.2

Cancel the common factors.

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Step 10.2.4.19.2.1

Factor $2$ out of $2$ .

$50 + 0 - (\frac{2 \cdot 50}{2 (1)} - 10 \cdot 10)$

Step 10.2.4.19.2.2

Cancel the common factor.

$50 + 0 - (\frac{2 \cdot 50}{2 \cdot 1} - 10 \cdot 10)$

Step 10.2.4.19.2.3

Rewrite the expression.

$50 + 0 - (\frac{50}{1} - 10 \cdot 10)$

Step 10.2.4.19.2.4

Divide $50$ by $1$ .

$50 + 0 - (50 - 10 \cdot 10)$

Step 10.2.4.20

Multiply $- 10$ by $10$ .

$50 + 0 - (50 - 100)$

Step 10.2.4.21

Subtract $100$ from $50$ .

$50 + 0 - - 50$

Step 10.2.4.22

Multiply $- 1$ by $- 50$ .

$50 + 0 + 50$

Step 10.2.4.23

Add $0$ and $50$ .

$50 + 50$

Step 10.2.4.24

Add $50$ and $50$ .

$100$

Step 11