Integration by Parts Integration by parts is an integration technique that comes from the product rule for derivatives. To simplify things while we introduce integration by parts. If u is a function, denote its derivative by D(u) and an antiderivative by I(u). Thus, for example, if u = 2x 2, then D(u) = 4x and I(u) = [If we wished, we could instead take I(u) = + 46, but we usually opt to take the simplest antiderivative.]

22 Integration by Parts Integration by parts If u and v are continuous functions of x, and u has a continuous derivative, then Quick Example

33 Example: Integration by Parts (Tabular Method) Calculate: Choose 1 function to be “u” and the other to be “v”. It is helpful to let “u” equal the easiest function to take the derivative of. Let u = x Use a table to calculate D(u) and I(v) Let v = e x The table is read as +x · e x −∫1 · e x dx

44 Example: Repeated Integration by Parts Solve:  x 2 e -x dx D I + -  X 2 e -x 2x -e -x = x 2(- e -x ) -  2x (-e -x ) dx The last integral is still a product. Continue the table alternating signs on the left. D I  X 2 e -x 2x -e -x 2 e -x 0 -e -x = x 2(- e -x ) - 2x(e -x ) + 2 (-e -x ) +C = -x 2 e -x - 2xe -x -2e -x + C = -e -x (x 2 + 2x + 2) + C To Summarize: Integrating a Polynomial Times a Function If one of the factors in the integrand is a polynomial and the other factor is a function that can be integrated repeatedly, put the polynomial in the D column and keep differentiating until you get zero. Then complete the I column to the same depth, and read off the answer.

Area Between Two Curves and Applications Area Between Two Graphs If f (x) ≥ g(x) for all x in [a, b] (so that the graph of f does not move below that of g), then the area of the region between the graphs of f and g and between x = a and x = b is given by

66 Example1: Area Between Two Curves Find the area between f (x) = –x 2 – 3x + 4 and g(x) = x 2 – 3x – 4 between x = –1 and x = 1

77 Example2: Area Between 2 Curves Find the Area between f (x) = | x | and g(x) = –| x – 1| over [–1, 2] Remember:  |x| dx = x |x| + C 2

88 General Process: Area Between Two Curves Finding the Area Between the Graphs of f (x) and g(x) 1. Find all points of intersection by solving f (x) = g(x) for x. This either determines the interval over which you will integrate or breaks up a given interval into regions between the intersection points. 2. Determine the area of each region you found by integrating the difference of the larger and the smaller function. (If you accidentally take the smaller minus the larger, the integral will give the negative of the area, so just take the absolute value.) 3. Add together the areas you found in step 2 to get the total area.

Averages To find the average of 20 numbers, add them up and divide by 20. More generally, the average, or mean, of the n numbers y 1, y 2, y 3,... y n, is the sum of the numbers divided by n. We write this average as (“y-bar”). Average, or Mean, of a Collection of Values The average of {0, 2, –1, 5} is

10 Example: Average Speed Over the course of 2 hours, my speed varied from 50 miles per hour to 60 miles per hour, following the function v(t) = t 2, 0 ≤ t ≤ 2. What was my average speed over those two hours? Recall : Average speed is total distance traveled divided by the time it took and we can find the distance traveled by integrating the speed: Distance traveled It took 2 hours to travel this distance, so the average speed was

11 Averages The average, or mean, of a function f (x) on an interval [a, b] is The average of f (x) = x on [1, 5] is

12 Interpreting the Average of a Function Geometrically Compare the graph of y = f (x) with the graph of y = 3, both over the interval [1, 5] We can find the area under the graph of f (x) = x by geometry or by calculus; it is 12. The area in the rectangle under y = 3 is also 12. Figure 8

13 Interpreting the Average of a Function Geometrically In general, the average of a positive function over the interval [a, b] gives the height of the rectangle over the interval [a, b] that has the same area as the area under the graph of f (x) The equality of these areas follows from the equation

14 Example: Average Balance A savings account at the People’s Credit Union pays 3% interest, compounded continuously, and at the end of the year you get a bonus of 1% of the average balance in the account during the year. If you deposit $10,000 at the beginning of the year, how much interest and how large a bonus will you get? Use the continuous compound interest formula to calculate the amount of money you have in the account at time t (in years) : A(t) = 10,000e 0.03t A(1) = $10, [Amount in account at end of 1 year] So you will have earned $ interest. To compute the bonus: find the average amount in the account, which is the average of A(t) over the interval [0, 1]. The bonus is 1% of this, or $ A= Pe rt