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John Abbott · Social Science · 201-SH3-AB

Calculus 2 tutoring for Social Science, 201-SH3-AB

Get one-to-one help setting up integrals and explaining what the answer means. Try the consumer-surplus example below, or bring your own question, even if you have not started it.

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Calculus 2 for Social Science at John Abbott

This course connects integration with area and economic applications. The short example below focuses on choosing bounds and interpreting a definite integral.

Topics to work on

  • Indefinite and definite integrals; consumer and producer surplus
  • Integration methods, including parts, partial fractions and tables
  • First-order separable differential equations
  • L’Hôpital’s rule, improper integrals, sequences and series convergence

If L’Hôpital’s rule is among your current topics, try this optional method-choice practice.

Before you calculate

Review the power rule for antiderivatives and solving a quadratic equation. In the example, first identify the quantity bought and the price paid. Those values determine the integration interval and the height you must subtract.

Steps worth checking

  • Finding the quantity from the price before choosing the upper bound
  • Distinguishing a price per kilogram, a total payment and consumer surplus

Worked example: find consumer surplus

In a simplified market model, qq is the quantity of a product bought during one market day, measured in kilograms. The demand curve D(q)=50−q220D(q)=50-\tfrac{q^2}{20}, in dollars per kilogram, gives buyers’ marginal willingness to pay: the rate at which the total amount they are willing to pay grows with quantity qq. The model applies for 0≤q≤300\le q\le30. All buyers pay p0=30p_0=30 dollars per kilogram, and all demand at that price is served. Find the quantity bought and the consumer surplus: total willingness to pay minus the amount actually paid.

Show a hint

Call the quantity bought QQ. First solve D(Q)=p0D(Q)=p_0 within the model’s domain. Consumer surplus adds the gap between the demand curve and the horizontal price line from q=0q=0 to q=Qq=Q.

Show the worked solution

  1. 1. Find the quantity and bounds

    At the quantity bought, marginal willingness to pay equals the price paid. Solve for QQ:

    50−Q220=30⟹Q2=400.50-\frac{Q^2}{20}=30\quad\Longrightarrow\quad Q^2=400.

    The nonnegative solution is Q=20Q=20 kilograms, which lies in the model’s domain. The integration interval is therefore [0,20][0,20].

  2. 2. Integrate the gap above the price

    Let SS be consumer surplus, in dollars. Subtract the uniform price before integrating:

    S=∫020(D(q)−30) dq=∫020(20−q220) dq=[20q−q360]020=8003.\begin{aligned}S&=\int_0^{20}\bigl(D(q)-30\bigr)\,\mathrm dq\\&=\int_0^{20}\left(20-\frac{q^2}{20}\right)\,\mathrm dq\\&=\left[20q-\frac{q^3}{60}\right]_0^{20}\\&=\frac{800}{3}.\end{aligned}

    Consumer surplus is about 266.67 dollars. Price is measured in dollars per kilogram and the quantity increment in kilograms, so the integral is measured in dollars. The demand curve is curved, so a triangle formula would not give the exact area.

  3. 3. Check the interpretation

    Buyers pay 30×20=60030\times20=600 dollars altogether. Adding this payment to the surplus gives total willingness to pay of 600+8003=26003600+\tfrac{800}{3}=\tfrac{2600}{3}, or about 866.67 dollars.

    The surplus measures buyers’ benefit above what they paid; it is not a refund. The 600-dollar payment is seller revenue. Seller profit would also require cost information.

Your turn: the price falls

Keep the same demand curve and domain, but let the uniform price fall to 5 dollars per kilogram. All demand at this price is served. (1) Find the new quantity QQ. (2) Write and evaluate the consumer-surplus integral. (3) Find the amount buyers pay and explain why it differs from the surplus.

Show a hint

Solve D(Q)=5D(Q)=5 before reusing any bound. Then integrate D(q)−5D(q)-5 over the quantities bought. For the amount paid, use price times quantity.

Check the three steps

  1. 1. Update the quantity

    50−Q220=5⟹Q2=900.50-\frac{Q^2}{20}=5\quad\Longrightarrow\quad Q^2=900.

    Take Q=30Q=30 kilograms. This is the upper endpoint of the stated domain, so the model covers this quantity.

  2. 2. Update both the bound and the price

    S=∫030(45−q220) dq=[45q−q360]030=1350−450=900.\begin{aligned}S&=\int_0^{30}\left(45-\frac{q^2}{20}\right)\,\mathrm dq\\&=\left[45q-\frac{q^3}{60}\right]_0^{30}\\&=1350-450=900.\end{aligned}

    Consumer surplus is 900 dollars. Keeping the old upper bound of 20 would leave out the additional quantity bought at the lower price.

  3. 3. Separate payment from surplus

    The amount paid is 5×30=1505\times30=150 dollars. Total willingness to pay is 150+900=1050150+900=1050 dollars. The payment is the amount transferred to the seller; the surplus is the benefit buyers retain above that payment.

If the setup was difficult, mark the horizontal price line, the upper bound and the gap being integrated before retrying. If the setup was right, check the antiderivative by differentiating it.

What to work through with a tutor

Bring a question, even if you do not know how to start. If you have already tried it, point out the first step you want to review. For an application, your tutor can help you define the quantities and units, choose the bounds, sketch the region and interpret the integral. Then try a changed price or interval yourself.

For a technique question, your tutor can help you compare methods and check your antiderivative by differentiation. For an improper integral or a series, they can help you write the required limit or choose a convergence test and explain why its conditions apply.

Build your review from your course outline

From your teacher’s announced topics, choose an integration-method question and an application. For the first, explain why the method fits and check the antiderivative by differentiating. For the application, sketch the region, choose the bounds and state the answer’s units and meaning. Redo a missed question without its solution. Follow your section’s instructions for permitted tools and assessed work.

Independent practice

Choose your next practice step

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