University Business Mathematics — Integral Calculus for Business Worksheet

All 12 questions

Each question targets one named concept from the sheet. Read them here, or print the PDF — it has working space under each one.

1. Q1Antiderivatives and Indefinite Integrals

   Find each indefinite integral, and state an interval on which your answer is valid. Check part (a) by differentiating.

   1. ${\displaystyle \int (12x^3-6\sqrt{x}+1)dx}$
   2. ${\displaystyle \int (\frac{7}{x}-3e^x)dx}$
   3. ${\displaystyle \int \frac{(2x+1{)}^2}{x^2}dx}$
2. Q2Antiderivatives and Indefinite Integrals

   A food truck sells $x$ meals a day, $0\le x\le 200$. Its marginal cost is $C^{\prime }(x)=12-0.04x$ dollars per meal and its fixed costs are $350 a day. Its marginal revenue is $R^{\prime }(x)=20-0.02x$ dollars per meal.

   1. Find the daily cost function $C(x)$.
   2. Find the revenue function $R(x)$, saying why its constant of integration is $0$, and deduce the price-demand function $p=D(x)$.
   3. Find the profit on a day when $100$ meals are sold.
3. Q3Integration by Substitution

   Evaluate each integral, stating the substitution you use.

   1. ${\displaystyle \int (x+1){(x^2+2x-5)}^{4}dx}$
   2. ${\displaystyle \int x^2{e}^{x^3-1}dx}$
   3. ${\displaystyle \int \frac{1}{x\ln x}dx}$, for $x>1$
4. Q4Integration by Substitution

   A furniture workshop makes $x$ dining chairs a month. Its marginal cost is $$C^{\prime }(x)=150+\frac{90}{\sqrt{2x+9}}\text{dollars per chair},$$ and its fixed costs are $2 000 a month, so $C(0)=2000$.

   1. Find the monthly cost function $C(x)$.
   2. Find the total cost of a month in which $20$ chairs are made.
5. Q5Riemann Sums and the Definite Integral

   A ceramics studio makes $x$ mugs a week. Its marginal cost is $$C^{\prime }(x)=0.01x^2-x+40\text{dollars per mug},0\le x\le 40,$$ which decreases on this interval. The increase in weekly cost when output goes from $0$ to $40$ mugs is ${\displaystyle {\int }_0^{40}{C}^{\prime }(x)dx}$.

   1. Split $[0,40]$ into $4$ subintervals of equal width. Write the right-endpoint sum $R_4$ in sigma notation, then evaluate it and the left-endpoint sum $L_4$.
   2. Which of the two overestimates the increase in cost, and why?
   3. The exact value of the integral is $\frac{3040}{3}$. Confirm that it lies between your two sums.
6. Q6The Fundamental Theorem and Net Change

   Evaluate each definite integral, showing the antiderivative you use.

   1. ${\displaystyle {\int }_0^2(x^3-3x+4)dx}$
   2. ${\displaystyle {\int }_1^9\frac{x+1}{\sqrt{x}}dx}$
   3. ${\displaystyle {\int }_0^2\frac{6x}{x^2+1}dx}$
7. Q7The Fundamental Theorem and Net Change

   A coffee roaster produces $x$ kilograms of coffee a week. Its marginal cost is $C^{\prime }(x)=9+0.0006x^2$ dollars per kilogram.

   1. Find the increase in weekly cost when production rises from $100$ to $200$ kilograms.
   2. Find the average value of the marginal cost over $100\le x\le 200$, and say in one sentence what it means for the roaster.
8. Q8Area Between Two Curves
   1. Find the area of the region enclosed by the parabolas $y=x^2-1$ and $y=7-x^2$.
   2. Find the area of the region between $y=e^x$ and $y=e^{-x}$ for $0\le x\le \ln 2$.
9. Q9Area Between Two Curves

   A print shop compares keeping its old press with leasing a new one. $t$ years from now, the old press would cost $8+0.6t^2$ thousand dollars a year to run and the new one $5+1.2t$ thousand dollars a year. Ignore interest.

   1. Show that the old press costs more to run at every time $t\ge 0$.
   2. Find the total running-cost saving of the new press over the next $5$ years, and say what region's area it is.
10. Q10Consumer and Producer Surplus

    In a regional market for wireless earbuds, the price-demand and price-supply functions are $$p=D(x)=\frac{400}{x+4},p=S(x)=4+6x,$$ where $x$ is the number of pairs in hundreds and $p$ the price in dollars. At the equilibrium $(\overline{x},\overline{p})$ the consumer surplus is ${\displaystyle {\int }_0^{\overline{x}}(D(x)-\overline{p})dx}$ and the producer surplus is ${\displaystyle {\int }_0^{\overline{x}}(\overline{p}-S(x))dx}$.

    1. Find the equilibrium quantity and price.
    2. Find the consumer surplus and the producer surplus, exactly and in dollars. Use $\ln 2.5\approx 0.9162907$.
11. Q11The Lorenz Curve and the Gini Index

    The distribution of annual pay among the employees of a retail chain is described by the Lorenz curve $$L(x)=0.3x+0.7x^2,0\le x\le 1,$$ meaning that the lowest-paid fraction $x$ of employees receives the fraction $L(x)$ of the total payroll. The Gini index is $G=2{\displaystyle {\int }_0^1(x-L(x))dx}$.

    1. Check that $L(0)=0$, $L(1)=1$ and $L(x)\le x$ on $[0,1]$.
    2. What percentage of the payroll goes to the lowest-paid half of employees?
    3. Find the Gini index.
12. Q12Synthesis — drawing on several topics in this unit

    In a harbour-cruise market, the price-demand function is $$p=D(x)=\frac{36}{\sqrt{2x+1}},$$ where $x$ is the number of tickets sold in a season, in thousands, and $p$ the ticket price in dollars. The market price settles where $4$ thousand tickets are sold.

    1. Find the market price.
    2. Find the consumer surplus ${\displaystyle {\int }_0^4(D(x)-\overline{p})dx}$ at that price, in dollars.

The 8 challenge problems for this topic are a separate, paid sheet and are not reproduced here.

How to do every concept on this sheet

This is the part worksheet sites usually leave out. Below is the actual reasoning behind each group of questions — not a full solution set, the decisions that get you to one. Read it before you start, or after you get stuck.

Antiderivatives, and where an answer is valid

An indefinite integral is a whole family of functions, all with the same derivative, and the + C is how you write the family down. Integration here means reading the differentiation rules backwards: raise a power by one and divide by the new power, remember that the power rule stops working at exactly one exponent, and recognise eˣ and 1/x as derivatives you already know.

Q1 asks for three of these and adds two demands. First, an interval on which each answer is valid: look at the integrand, not the answer, and ask where it is defined. A square root, a 1/x and a division by x² each rule something out, and an interval must stay on one side of any point that is excluded. Second, part (a) is to be checked by differentiating — which is the check for every indefinite integral you will ever write.

A cost function from its marginal cost

Every business reading in this chapter starts from one sentence: the marginal cost is the derivative of the cost, so the cost is an antiderivative of the marginal cost. The integral alone gives a family; the business situation picks out one member of it.

Q2 does this twice, for a food truck. Part (a) builds the daily cost from its marginal cost and fixed costs. Part (b) builds revenue the same way, and asks you to say why its constant is what it is — a reason from the situation, not from algebra — and then to recover the price-demand function. Revenue is price times quantity, R = xp, so once R(x) is known the price is what you get by dividing by x. Part (c) needs both functions: profit is revenue minus cost, evaluated at the given number of meals.

Integration by substitution

Substitution is the chain rule read backwards. It works when the integrand contains a function and, as a factor, the derivative of that function — up to a constant multiple, which you can adjust for. Name the inside function u, and the integral becomes one of the basic forms in u.

Q3 gives three integrals and asks you to state the substitution you use, so the choice is part of what is marked. Part (a) is a power of a polynomial times a factor that is a multiple of its derivative; part (b) is an exponential whose exponent is a polynomial; part (c) mixes a logarithm with 1/x, and the stated domain x > 1 is there to keep the logarithm, and the answer, meaningful.

Q4 puts substitution in a furniture workshop. The marginal cost has a square root of a linear expression in a denominator, which is a power in disguise: rewrite it with a negative fractional exponent before choosing u.

Riemann sums, and which way an estimate leans

A definite integral of a marginal cost over a range of output is the increase in cost across that range, and before the Fundamental Theorem the way to reach it is to add up rectangles. Split the interval into n pieces of equal width Δx = (b − a)/n, take one height per piece, and add height times width.

Q5 does this for a ceramics studio. Part (a) wants the right-endpoint sum written in sigma notation first and then evaluated, together with the left-endpoint sum. List the five endpoints before you compute anything: the left sum uses the first four, the right sum the last four, and reaching for the wrong list is the usual slip.

Part (c) gives the exact value of the integral and asks you to confirm it lies between your two sums. That is the check that your part (b) reasoning and your arithmetic agree.

The Fundamental Theorem and net change

The theorem replaces the limit of sums with one subtraction: find any antiderivative F of f, and the integral over [a, b] is F(b) − F(a). No + C is needed, because it would cancel.

Q6 asks for three definite integrals, showing the antiderivative each time. Part (a) is a polynomial. Part (b) is a fraction over √x: split it into two powers of x first, exactly as in Q1(c). Part (c) needs a substitution, and a definite integral gives you a choice — change the limits to values of u and never return to x, or return to x and use the original limits. Do one or the other; mixing them, with u-limits on an x-antiderivative, is the error.

Q7 is the business reading. For the coffee roaster, the integral of the marginal cost between two production levels is the extra weekly cost of producing the extra coffee — no fixed cost is needed, because it cancels in the subtraction. Part (b) asks for the average value of the marginal cost over the same range.

Area between two curves

The area between two curves is the integral of the top curve minus the bottom one. Everything that can go wrong goes wrong in deciding the interval or deciding which curve is on top.

Q8(a) is an enclosed region between two parabolas, so its limits come from solving for the points of intersection. Q8(b) gives the interval, with an upper limit written as a logarithm: evaluate eˣ and e⁻ˣ at that limit with the law e^(ln a) = a rather than with a calculator, and test a point to find which exponential is on top.

Q9 is the same idea for a print shop comparing two presses. Part (a) asks you to show that one running cost is larger at every time t ≥ 0: subtract the two, and show the difference is positive for every t — a quadratic that never reaches zero is the tool, and either its discriminant or completing the square settles it. Part (b) asks for the saving over a period and for the region whose area it is; that region is bounded by the two cost curves and two vertical lines, and the answer should say which.

Consumer and producer surplus

At the market equilibrium, supply meets demand at a quantity x̄ and a price p̄. Consumers who would have paid more than p̄ keep the difference, and the total of what they keep is the area between the demand curve and the horizontal line at p̄. Producers who would have sold for less keep the area between that line and the supply curve.

Q10 sets up a market for wireless earbuds. Part (a) finds the equilibrium by setting D(x) = S(x): clear the denominator, and the equation becomes a quadratic. A quantity sold cannot be negative, so only one root is the equilibrium. The price then comes from either function — compute it from both, as a check.

The Lorenz curve and the Gini index

A Lorenz curve L(x) says what fraction of the total the lowest-paid fraction x of people receives. Perfect equality is the line L(x) = x, and the further a curve sags below that line the more unequal the distribution. The Gini index doubles the region that lies under the line of equality and over the Lorenz curve, which puts it between 0 (perfect equality) and 1.

Q11 describes a retail chain's payroll. Part (a) checks that the given L is a legitimate Lorenz curve: evaluate at both ends, then show L(x) ≤ x on [0, 1] by writing x − L(x) as one polynomial, factoring it, and reading its sign from the factors. Part (b) is a reading of the definition — the lowest-paid half means x = 0.5 — converted to a percentage. Part (c) is the integral in the formula given, a polynomial integral with no technique needed.

The synthesis question

Q12 brings the set together in a harbour-cruise market with a demand function that is a constant over a square root. Part (a) reads the market price off the demand function at the given quantity. Part (b) is the consumer surplus integral — and D(x) is exactly the kind of function Q4 taught you to integrate: rewrite the root as a power, substitute for the linear expression under it, and either change the limits or return to x, as in Q6(c). The quantity is in thousands of tickets, so the last step, as in Q10, is a conversion to dollars.

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Getting the most out of it

Differentiate every indefinite integral you write

It takes a few seconds and it catches a wrong power, a lost constant from a substitution and a sign error all at once. Q1 asks for it once; make it a habit on every question.

Pin the constant with the business, not with a guess

Before integrating a marginal function, write down the one value you know — the fixed cost, the revenue at zero sales — and after integrating, substitute it into the whole antiderivative. Q2 and Q4 are both decided at this step.

Draw the region before you write an integral

For an area, a surplus or a Gini index, a rough sketch tells you the limits and which curve is on top, and it tells you whether your final number is plausible. Three minutes on the sketch saves an integral of the wrong difference.

Finish with the units

Every applied question here measures quantity in some unit — meals, chairs, kilograms, hundreds of pairs, thousands of tickets — and wants a total in dollars. State the units of the number you found before you decide it is the answer.

Common questions