Calculus Examples

s (q) = 5.2 q

$s (q) = 5.2 q$ ,

(2, 130)

$(2, 130)$

Step 1

Set up the producer surplus

q_{eq} p_{eq} - \int_{0}^{q_{eq}} s (q) d q

$q_{eq} p_{eq} - \int_{0}^{q_{eq}} s (q) d q$ where

q_{eq}

$q_{eq}$ is the equilibrium quantity and

p_{eq}

$p_{eq}$ is the equilibrium price.

2 \cdot 130 - \int_{0}^{2} 5.2 q d q

$2 \cdot 130 - \int_{0}^{2} 5.2 q d q$

Step 2

Evaluate the integral and simplify.

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Step 2.1

Multiply

2

$2$ by

130

$130$ .

260 - \int_{0}^{2} 5.2 q d q

$260 - \int_{0}^{2} 5.2 q d q$

Step 2.2

Since

5.2

$5.2$ is constant with respect to

q

$q$ , move

5.2

$5.2$ out of the integral.

260 - (5.2 \int_{0}^{2} q d q)

$260 - (5.2 \int_{0}^{2} q d q)$

Step 2.3

Multiply

5.2

$5.2$ by

- 1

$- 1$ .

260 - 5.2 \int_{0}^{2} q d q

$260 - 5.2 \int_{0}^{2} q d q$

Step 2.4

By the Power Rule, the integral of

q

$q$ with respect to

q

$q$ is

\frac{1}{2} q^{2}

$\frac{1}{2} q^{2}$ .

260 - 5.2 (\frac{1}{2} q^{2}]_{0}^{2})

$260 - 5.2 (\frac{1}{2} q^{2}]_{0}^{2})$

Step 2.5

Simplify the answer.

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Step 2.5.1

Combine

\frac{1}{2}

$\frac{1}{2}$ and

q^{2}

$q^{2}$ .

260 - 5.2 (\frac{q^{2}}{2}]_{0}^{2})

$260 - 5.2 (\frac{q^{2}}{2}]_{0}^{2})$

Step 2.5.2

Substitute and simplify.

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Step 2.5.2.1

Evaluate

\frac{q^{2}}{2}

$\frac{q^{2}}{2}$ at

2

$2$ and at

0

$0$ .

260 - 5.2 ((\frac{2^{2}}{2}) - \frac{0^{2}}{2})

$260 - 5.2 ((\frac{2^{2}}{2}) - \frac{0^{2}}{2})$

Step 2.5.2.2

Simplify.

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Step 2.5.2.2.1

Raise

2

$2$ to the power of

2

$2$ .

260 - 5.2 (\frac{4}{2} - \frac{0^{2}}{2})

$260 - 5.2 (\frac{4}{2} - \frac{0^{2}}{2})$

Step 2.5.2.2.2

Cancel the common factor of

4

$4$ and

2

$2$ .

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Step 2.5.2.2.2.1

Factor

2

$2$ out of

4

$4$ .

260 - 5.2 (\frac{2 \cdot 2}{2} - \frac{0^{2}}{2})

$260 - 5.2 (\frac{2 \cdot 2}{2} - \frac{0^{2}}{2})$

Step 2.5.2.2.2.2

Cancel the common factors.

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Step 2.5.2.2.2.2.1

Factor

2

$2$ out of

2

$2$ .

260 - 5.2 (\frac{2 \cdot 2}{2 (1)} - \frac{0^{2}}{2})

$260 - 5.2 (\frac{2 \cdot 2}{2 (1)} - \frac{0^{2}}{2})$

Step 2.5.2.2.2.2.2

Cancel the common factor.

260 - 5.2 (\frac{2 \cdot 2}{2 \cdot 1} - \frac{0^{2}}{2})

$260 - 5.2 (\frac{2 \cdot 2}{2 \cdot 1} - \frac{0^{2}}{2})$

Step 2.5.2.2.2.2.3

Rewrite the expression.

260 - 5.2 (\frac{2}{1} - \frac{0^{2}}{2})

$260 - 5.2 (\frac{2}{1} - \frac{0^{2}}{2})$

Step 2.5.2.2.2.2.4

Divide

2

$2$ by

1

$1$ .

260 - 5.2 (2 - \frac{0^{2}}{2})

$260 - 5.2 (2 - \frac{0^{2}}{2})$

260 - 5.2 (2 - \frac{0^{2}}{2})

$260 - 5.2 (2 - \frac{0^{2}}{2})$

260 - 5.2 (2 - \frac{0^{2}}{2})

$260 - 5.2 (2 - \frac{0^{2}}{2})$

Step 2.5.2.2.3

Raising

0

$0$ to any positive power yields

0

$0$ .

260 - 5.2 (2 - \frac{0}{2})

$260 - 5.2 (2 - \frac{0}{2})$

Step 2.5.2.2.4

Cancel the common factor of

0

$0$ and

2

$2$ .

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Step 2.5.2.2.4.1

Factor

2

$2$ out of

0

$0$ .

260 - 5.2 (2 - \frac{2 (0)}{2})

$260 - 5.2 (2 - \frac{2 (0)}{2})$

Step 2.5.2.2.4.2

Cancel the common factors.

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Step 2.5.2.2.4.2.1

Factor

2

$2$ out of

2

$2$ .

260 - 5.2 (2 - \frac{2 \cdot 0}{2 \cdot 1})

$260 - 5.2 (2 - \frac{2 \cdot 0}{2 \cdot 1})$

Step 2.5.2.2.4.2.2

Cancel the common factor.

260 - 5.2 (2 - \frac{2 \cdot 0}{2 \cdot 1})

$260 - 5.2 (2 - \frac{2 \cdot 0}{2 \cdot 1})$

Step 2.5.2.2.4.2.3

Rewrite the expression.

260 - 5.2 (2 - \frac{0}{1})

$260 - 5.2 (2 - \frac{0}{1})$

Step 2.5.2.2.4.2.4

Divide

0

$0$ by

1

$1$ .

260 - 5.2 (2 - 0)

$260 - 5.2 (2 - 0)$

260 - 5.2 (2 - 0)

$260 - 5.2 (2 - 0)$

260 - 5.2 (2 - 0)

$260 - 5.2 (2 - 0)$

Step 2.5.2.2.5

Multiply

- 1

$- 1$ by

0

$0$ .

260 - 5.2 (2 + 0)

$260 - 5.2 (2 + 0)$

Step 2.5.2.2.6

Add

2

$2$ and

0

$0$ .

260 - 5.2 \cdot 2

$260 - 5.2 \cdot 2$

Step 2.5.2.2.7

Multiply

- 5.2

$- 5.2$ by

2

$2$ .

260 - 10.4

$260 - 10.4$

Step 2.5.2.2.8

Subtract

10.4

$10.4$ from

260

$260$ .

249.6

$249.6$

249.6

$249.6$

249.6

$249.6$

249.6

$249.6$

249.6

$249.6$

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