Chapter 6 Review
Chapter 6: Using the integral
- SECTION 6.1: Average value
- Suggested Problems: #2, 3, 4, 9, 10, 13, 16.
- If f(x) is a function that is continuous over the interval [a, b], then the average value of f(x) from x = a to x = b is given by (b - a)-1
f(x) dx.
- The average value of f(x) over the interval [a, b] yields the height of the rectangle having width (b - a) that has the same (signed) area as that of the region between the curve y = f(x) and the x-axis over the interval [a, b].
- Be able to find the average value of a function f(x) over an interval [a, b], whether f(x) is expressed as a formula or as a graph.
- SECTION 6.2: Consumer and producer surplus
- Suggested Problems: #1, 3, 4, 10.
- Recall that the supply curve shows the quantity of an item that producers will supply at different price levels, and the demand curve is the quantity of the item that consumers will buy at different price levels.
- It is assumed that the market will settle on the equilibrium price p* and equilibrium quantity q* where the supply and the demand curves cross. This is the equilibrium point at which supply equals demand. The quantity q* of the item will be produced and sold for the price p* each.
- The equilibrium point represents the collective behavior of the consumers and the producers, but will not necessarily represent the behavior of each individual---some consumers will be willing to pay more for a given item, and some producers will be willing to sell an item for less.
- Consumer Surplus is a measure of the consumers' gain from trade. It is the total amount gained by consumers by purchasing an item at the current price, rather than at the price at which they would have been willing to pay.
- Producer Surplus is a measure of the producers' gain from trade. It is the total amount gained by producers by selling an item at the current price, rather than at the price at which they would have been willing to sell.
- The total gains from trade are given by the sum of the consumer surplus and the producer surplus.
- Note: in the above definitions, the current price is assumed to be the equilibrium price in the absence of price controls.
- Given a supply curve p = S(q) and a demand curve p = D(q), be able to find the equilibrium price p* and the equilibrium quantity q*.
- Given a supply curve p = S(q) and a demand curve p = D(q), the consumer surplus is given by
-p* q* + 
D(q) dq,
which is the area under the demand curve p = D(q) and above the horizontal line p = p* from q = 0 to q = q*.
- Given a supply curve p = S(q) and a demand curve p = D(q), the producer surplus is given by
p* q* - S(q) dq,
which is the area above the supply curve p = S(q) and below the horizontal line p = p* from q = 0 to q = q*.
- The total gains from trade is then
D(q) dq - S(q) dq,
which consists of the total area below the demand curve p = D(q) and above the supply curve p = S(q) from q = 0 to q = q*.
- Know what parts of the graph of a supply curve and a demand curve yields the consumer surplus, the producer surplus, and the total gains from trade.
- Be able to find the consumer surplus, the producer surplus, and the total gains from trade given either formulas for the supply and demand curves, or graphs of these curves.
- SECTION 6.3: Present and future value
- Suggested Problems: #2, 3, 8, 10, 11.
- Payments made by or to an individual are usually made at specific points in time, with a nonzero gap of time between each payment. The paymets are then considered to be made at discrete points in time.
- The revenue for a huge corporation comes in all of the time, and so we describe the revenue in terms of a rate at which it is earned. This rate is known as an income stream. Since there is a significant rate of revenue at all points in time, the income stream is described as a continuous function S(t), in the units dollars per year, with time t in years from the present.
- In Section 1.9, we considered the present value and the future value of a
discrete payment. We can also do this with a continuous income stream.
But in this case we assume that interest is compounded continuously.
- The present value of an income stream (at t = 0) described by the rate of S(t) dollars per year for the period of time from now (t = 0) until M years in the future (t = M), is given by the definite integral
Present value = 
S(t) e-rt dt,
where r is the continuous interest rate at which the money from the income stream is invested as soon as it is received.
- The future value of an income stream (at t = M) described by the rate of S(t) dollars per year for the period of time from now (t = 0) until M years in the future (t = M), is given by
Future value = (Present value) · erM,
where r is the continuous interest rate at which the money from the income stream is invested as soon as it is received.
- The total income received from an income stream described by the rate of S(t) dollars per year for the period of time from now (t = 0) until M years in the future (t = M), is given by
S(t) dt.
- SECTION 6.5: Antiderivatives
- Suggested Problems: #30, 31, 39, 44, 45, 46.
- If the derivative of the function F(x) is f(x), so that F
(x) = f(x), then F(x) is said to be an antiderivative of f(x).
- Every function that has an antiderivative must have more than one antiderivative since
(d / dx) (F(x) + C) = F(x)
for any constant C.
- Any two functions that have the same derivative must differ by a constant.
- It can be shown that if F(x) is an antiderivative of some function f(x), then any antiderivative of f(x) can be written in the form F(x) + C. Thus, if we let C range over all constants, then the expression F(x) + C will represent all of the possible antiderivatives of f(x).
- The indefinite integral of f(x), written
f(x) dx, represents the collection of all antiderivatives of a function f(x). Thus, if F(x) is one antiderivative of f(x) (so that F(x) = f(x)), then the collection of all antiderivatives of f(x) is given by f(x) dx, so that
f(x) dx = F(x) + C,
where C is an arbitrary constant.
- Be familiar with the formulas for antiderivatives, and be able to use them to find the antiderivative of a given function.
- Formulas for antiderivatives.
- The antiderivative of a power of x when the power is not -1:
xn dx = (1/(n+1)) xn+1 + C, provided n 
-1.
- The antiderivative of a power of x when the power is -1:
x-1 dx = ln|x| + C
- The antiderivative of a constant k:
k dx = kx + C
- The antiderivative of a sum or a difference of functions:
( f(x) ± g(x) ) dx = f(x) dx ± g(x) dx
- The antiderivative of a constant times a function:
a f(x) dx = a f(x) dx
- The antiderivative of the exponential function:
ex dx = ex + C
More generally:
ekx dx = (1 / k) ekx + C
- SECTION 6.6: Using antiderivatives to find definite integrals
- Suggested Problems: #5, 6, 7, 8, 9, 10, 26.
- The Fundamental Theorem of Calculus: If F(x) is continuous on an interval containing a and b, then
F(x) dx = F(b) - F(a).
Thus, the definite integral, from a to b, of a function is the antiderivative of that function evaluated at b minus the antiderivative of that function evaluated at a.
- Let us denote
F(x) 
= F(b) - F(a).
Then the Fundamental Theorem of Calculus can be restated as
F(x) dx = F(x) .
- Note that for any constant C,
(F(x) + C) = (F(b) + C) - (F(a) + C) = F(b) - F(a) = F(x) ,
and so including the constant of integration with an antiderivative is not necessary when evaluating a definite integral.
- Be able to find the exact value of a definite integral (for certain functions) by means of finding the antiderivative of the integrand, and then by applying the Fundamental Theorem of Calculus.
- A definite integral f(x) dx is said to be improper if either one (or both) of the limits of integration is infinite, or if the integrand f(x) is unbounded on the interval [a, b].
- Be able to work with improper integrals.
Last modified: Fri May 03 2002