Calculus Examples

$s (q) = 5.2 q$ , $(2, 130)$

Step 1

Set up the producer surplus $q_{eq} p_{eq} - \int_{0}^{q_{eq}} s (q) d q$ where $q_{eq}$ is the equilibrium quantity and $p_{eq}$ is the equilibrium price.

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

Step 2

Evaluate the integral and simplify.

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Step 2.1

Multiply $2$ by $130$ .

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

Step 2.2

Since $5.2$ is constant with respect to $q$ , move $5.2$ out of the integral.

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

Step 2.3

Multiply $5.2$ by $- 1$ .

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

Step 2.4

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

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

Step 2.5

Simplify the answer.

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Step 2.5.1

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

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

Step 2.5.2

Substitute and simplify.

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Step 2.5.2.1

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

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

Step 2.5.2.2

Simplify.

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Step 2.5.2.2.1

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

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

Step 2.5.2.2.2

Cancel the common factor of $4$ and $2$ .

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Step 2.5.2.2.2.1

Factor $2$ out of $4$ .

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

Step 2.5.2.2.2.2

Cancel the common factors.

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Step 2.5.2.2.2.2.1

Factor $2$ out of $2$ .

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

Step 2.5.2.2.2.2.2

Cancel the common factor.

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

Step 2.5.2.2.2.2.3

Rewrite the expression.

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

Step 2.5.2.2.2.2.4

Divide $2$ by $1$ .

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

Step 2.5.2.2.3

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

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

Step 2.5.2.2.4

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

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Step 2.5.2.2.4.1

Factor $2$ out of $0$ .

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

Step 2.5.2.2.4.2

Cancel the common factors.

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Step 2.5.2.2.4.2.1

Factor $2$ out of $2$ .

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

Step 2.5.2.2.4.2.2

Cancel the common factor.

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

Step 2.5.2.2.4.2.3

Rewrite the expression.

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

Step 2.5.2.2.4.2.4

Divide $0$ by $1$ .

$260 - 5.2 (2 - 0)$

Step 2.5.2.2.5

Multiply $- 1$ by $0$ .

$260 - 5.2 (2 + 0)$

Step 2.5.2.2.6

Add $2$ and $0$ .

$260 - 5.2 \cdot 2$

Step 2.5.2.2.7

Multiply $- 5.2$ by $2$ .

$260 - 10.4$

Step 2.5.2.2.8

Subtract $10.4$ from $260$ .

$249.6$

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