Calculus Examples

d (q) = 300 - 5 q

$d (q) = 300 - 5 q$ ,

(15, 225)

$(15, 225)$

Step 1

Set up the consumer surplus

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

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

q_{eq}

$q_{eq}$ is the equilibrium quantity and

p_{eq}

$p_{eq}$ is the equilibrium price.

\int_{0}^{15} 300 - 5 q d q - 15 \cdot 225

$\int_{0}^{15} 300 - 5 q d q - 15 \cdot 225$

Step 2

Evaluate the integral and simplify.

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Step 2.1

Multiply

- 15

$- 15$ by

225

$225$ .

\int_{0}^{15} 300 - 5 q d q - 3375

$\int_{0}^{15} 300 - 5 q d q - 3375$

Step 2.2

Split the single integral into multiple integrals.

\int_{0}^{15} 300 d q + \int_{0}^{15} - 5 q d q - 3375

$\int_{0}^{15} 300 d q + \int_{0}^{15} - 5 q d q - 3375$

Step 2.3

Apply the constant rule.

300 q]_{0}^{15} + \int_{0}^{15} - 5 q d q - 3375

$300 q]_{0}^{15} + \int_{0}^{15} - 5 q d q - 3375$

Step 2.4

Since

- 5

$- 5$ is constant with respect to

q

$q$ , move

- 5

$- 5$ out of the integral.

300 q]_{0}^{15} - 5 \int_{0}^{15} q d q - 3375

$300 q]_{0}^{15} - 5 \int_{0}^{15} q d q - 3375$

Step 2.5

By the Power Rule, the integral of

q

$q$ with respect to

q

$q$ is

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

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

300 q]_{0}^{15} - 5 (\frac{1}{2} q^{2}]_{0}^{15}) - 3375

$300 q]_{0}^{15} - 5 (\frac{1}{2} q^{2}]_{0}^{15}) - 3375$

Step 2.6

Simplify the answer.

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Step 2.6.1

Combine

\frac{1}{2}

$\frac{1}{2}$ and

q^{2}

$q^{2}$ .

300 q]_{0}^{15} - 5 (\frac{q^{2}}{2}]_{0}^{15}) - 3375

$300 q]_{0}^{15} - 5 (\frac{q^{2}}{2}]_{0}^{15}) - 3375$

Step 2.6.2

Substitute and simplify.

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Step 2.6.2.1

Evaluate

300 q

$300 q$ at

15

$15$ and at

0

$0$ .

(300 \cdot 15) - 300 \cdot 0 - 5 (\frac{q^{2}}{2}]_{0}^{15}) - 3375

$(300 \cdot 15) - 300 \cdot 0 - 5 (\frac{q^{2}}{2}]_{0}^{15}) - 3375$

Step 2.6.2.2

Evaluate

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

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

15

$15$ and at

0

$0$ .

300 \cdot 15 - 300 \cdot 0 - 5 (\frac{15^{2}}{2} - \frac{0^{2}}{2}) - 3375

$300 \cdot 15 - 300 \cdot 0 - 5 (\frac{15^{2}}{2} - \frac{0^{2}}{2}) - 3375$

Step 2.6.2.3

Simplify.

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Step 2.6.2.3.1

Multiply

300

$300$ by

15

$15$ .

4500 - 300 \cdot 0 - 5 (\frac{15^{2}}{2} - \frac{0^{2}}{2}) - 3375

$4500 - 300 \cdot 0 - 5 (\frac{15^{2}}{2} - \frac{0^{2}}{2}) - 3375$

Step 2.6.2.3.2

Multiply

- 300

$- 300$ by

0

$0$ .

4500 + 0 - 5 (\frac{15^{2}}{2} - \frac{0^{2}}{2}) - 3375

$4500 + 0 - 5 (\frac{15^{2}}{2} - \frac{0^{2}}{2}) - 3375$

Step 2.6.2.3.3

Add

4500

$4500$ and

0

$0$ .

4500 - 5 (\frac{15^{2}}{2} - \frac{0^{2}}{2}) - 3375

$4500 - 5 (\frac{15^{2}}{2} - \frac{0^{2}}{2}) - 3375$

Step 2.6.2.3.4

Raise

15

$15$ to the power of

2

$2$ .

4500 - 5 (\frac{225}{2} - \frac{0^{2}}{2}) - 3375

$4500 - 5 (\frac{225}{2} - \frac{0^{2}}{2}) - 3375$

Step 2.6.2.3.5

Raising

0

$0$ to any positive power yields

0

$0$ .

4500 - 5 (\frac{225}{2} - \frac{0}{2}) - 3375

$4500 - 5 (\frac{225}{2} - \frac{0}{2}) - 3375$

Step 2.6.2.3.6

Cancel the common factor of

0

$0$ and

2

$2$ .

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Step 2.6.2.3.6.1

Factor

2

$2$ out of

0

$0$ .

4500 - 5 (\frac{225}{2} - \frac{2 (0)}{2}) - 3375

$4500 - 5 (\frac{225}{2} - \frac{2 (0)}{2}) - 3375$

Step 2.6.2.3.6.2

Cancel the common factors.

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Step 2.6.2.3.6.2.1

Factor

2

$2$ out of

2

$2$ .

4500 - 5 (\frac{225}{2} - \frac{2 \cdot 0}{2 \cdot 1}) - 3375

$4500 - 5 (\frac{225}{2} - \frac{2 \cdot 0}{2 \cdot 1}) - 3375$

Step 2.6.2.3.6.2.2

Cancel the common factor.

$4500 - 5 (\frac{225}{2} - \frac{2 \cdot 0}{2 \cdot 1}) - 3375$

Step 2.6.2.3.6.2.3

Rewrite the expression.

$4500 - 5 (\frac{225}{2} - \frac{0}{1}) - 3375$

Step 2.6.2.3.6.2.4

Divide

$0$ by

$1$ .

$4500 - 5 (\frac{225}{2} - 0) - 3375$

Step 2.6.2.3.7

Multiply

$- 1$ by

$0$ .

$4500 - 5 (\frac{225}{2} + 0) - 3375$

Step 2.6.2.3.8

Add

$\frac{225}{2}$ and

$0$ .

$4500 - 5 (\frac{225}{2}) - 3375$

Step 2.6.2.3.9

Combine

$- 5$ and

$\frac{225}{2}$ .

$4500 + \frac{- 5 \cdot 225}{2} - 3375$

Step 2.6.2.3.10

Multiply

$- 5$ by

$225$ .

$4500 + \frac{- 1125}{2} - 3375$

Step 2.6.2.3.11

Move the negative in front of the fraction.

$4500 - \frac{1125}{2} - 3375$

Step 2.6.2.3.12

To write

$4500$ as a fraction with a common denominator, multiply by

$\frac{2}{2}$ .

$4500 \cdot \frac{2}{2} - \frac{1125}{2} - 3375$

Step 2.6.2.3.13

Combine

$4500$ and

$\frac{2}{2}$ .

$\frac{4500 \cdot 2}{2} - \frac{1125}{2} - 3375$

Step 2.6.2.3.14

Combine the numerators over the common denominator.

$\frac{4500 \cdot 2 - 1125}{2} - 3375$

Step 2.6.2.3.15

Simplify the numerator.

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Step 2.6.2.3.15.1

Multiply

$4500$ by

$2$ .

$\frac{9000 - 1125}{2} - 3375$

Step 2.6.2.3.15.2

Subtract

$1125$ from

$9000$ .

$\frac{7875}{2} - 3375$

Step 2.6.2.3.16

To write

$- 3375$ as a fraction with a common denominator, multiply by

$\frac{2}{2}$ .

$\frac{7875}{2} - 3375 \cdot \frac{2}{2}$

Step 2.6.2.3.17

Combine

$- 3375$ and

$\frac{2}{2}$ .

$\frac{7875}{2} + \frac{- 3375 \cdot 2}{2}$

Step 2.6.2.3.18

Combine the numerators over the common denominator.

$\frac{7875 - 3375 \cdot 2}{2}$

Step 2.6.2.3.19

Simplify the numerator.

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Step 2.6.2.3.19.1

Multiply

$- 3375$ by

$2$ .

$\frac{7875 - 6750}{2}$

Step 2.6.2.3.19.2

Subtract

$6750$ from

$7875$ .

$\frac{1125}{2}$

Step 2.7

Divide

$1125$ by

$2$ .

$562.5$

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