Marianopolis College · 201-SH3-MS

Marianopolis 201-SH3-MS: Integral Calculus in Social Science Tutoring

In 201-SH3-MS, the main challenge is often deciding what an integral should add up. An area between economic curves, a probability and an income-distribution measure can use similar calculations while representing different quantities. Nablio helps Marianopolis students choose the integrand, limits and units before integrating. The original example below turns a market diagram into consumer and producer surplus.

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Build an integral that answers the question

Definite integrals and antiderivatives: distinguish an accumulated value from a family of functions.
Integration methods and improper integrals: choose a method and check whether a limiting value exists.
Consumer and producer surplus: identify the correct curve, market price and quantity interval.
Probability and inequality measures: interpret density areas and the area associated with a Lorenz curve before calculating.

Draw and label the region first

Find any intersection needed for the integration limits. For an area between curves, identify which function is above the other throughout the interval. In a surplus calculation, a price difference measured in dollars per unit is multiplied by a small quantity interval, giving dollars. For probability, integrate a nonnegative density over the event; its total area across all possible values must equal one. These checks expose many setup errors before an antiderivative is needed.

Worked example: two different areas at market equilibrium

An original continuous market model has demand price $D(q)=80-2q$ and supply price $S(q)=20+q$ , both in dollars per unit. Find the equilibrium quantity and price, then the consumer and producer surplus. Treat the curves as valid over the relevant nonnegative quantities.

At equilibrium, demand and supply prices agree: $80-2q=20+q$ . Thus $q_*=20$ units and $p_*=40$ dollars per unit.
For $0\le q<20$ , the demand price is above the market price; at $q=20$ , the two prices are equal. Consumer surplus is $\begin{aligned}CS&=\int_0^{20}(40-2q)\,dq\\&=[40q-q^2]_0^{20}=400.\end{aligned}$ It is 400 dollars, not 400 dollars per unit.
For producers, subtract the supply price from the market price: $\begin{aligned}PS&=\int_0^{20}(20-q)\,dq\\&=[20q-q^2/2]_0^{20}=200.\end{aligned}$ Producer surplus is 200 dollars.
Check the geometry: the two regions are triangles with base 20. Their heights are 40 and 20, giving areas $20(40)/2=400$ and $20(20)/2=200$ .

The 800 dollars of market expenditure, found from price times quantity, is neither surplus. Each surplus measures a different area relative to the common market price.

Try a probability integral

A nonnegative continuous variable X has density $f(x)=kx$ for $0\le x\le2$ , and zero elsewhere. Find k so that this is a valid density, then calculate $P(1\le X\le2)$ .

Show the explained answer

The total area must be one: $\int_0^2 kx\,dx=2k=1$ , so $k=1/2$ . The requested probability is $\int_1^2 \frac{x}{2}\,dx=\left[\frac{x^2}{4}\right]_1^2=\frac34.$ Integrating over the event gives 0.75; the density’s value at one point is not itself a probability.

Original Nablio example and practice, independent of the college. These are not course assessment questions. Follow your current section’s outline for the material to review.

Check the method, then the interval

If you chose the correct surplus region or probability event but stalled at the antiderivative, start with simplifying before choosing an integration technique. The examples above use polynomial integrands, so basic antiderivative rules are enough; move to substitution or parts only when the new integrand calls for them.
If a later model has an infinite integration bound or an integrand that becomes unbounded, use the definition of an improper integral before calculating. The density above has finite support and is bounded, so its normalization does not need an improper limit.

For a density, keep checking nonnegativity, the stated support and total probability one. A method that finds an antiderivative does not by itself establish a valid probability model.

Set up an integral with your tutor

Bring the question and your current course outline, if you have them. You can draw the graph and set up the integral with your tutor, or review any work you have started. We can work on setting limits, choosing a technique or explaining why an answer has the expected sign and units. If your question concerns improper integrals or the Gini index, include the precise model so the session follows the definitions used in your course.

We use permitted practice without completing graded assignments or exams on your behalf.

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