Answer: (a) Use six rectangles to find estimates of each type for the area under the

(a) By reading values from the given graph of f, use four rectangles to find a lower estimate and an upper estimate for the area under the given graph of f from x 0 to x 8. In each case sketch the rectangles that you use. (b) Find new estimates using eight rectangles in each case

(a) Use six rectangles to find estimates of each type for the area under the given graph of f from x 0 to x 12. (i) L6 (sample points are left endpoints) (ii) R6 (sample points are right endpoints) (iii) M6 (sample points are midpoints) (b) Is L6 an underestimate or overestimate of the true area? (c) Is R6 an underestimate or overestimate of the true area? (d) Which of the numbers L6, R6, or M6 gives the best estimate? Explain

. (a) Estimate the area under the graph of fsxd cos x from x 0 to x y2 using four approximating rectangles and right endpoints. Sketch the graph and the rectangles. Is your estimate an underestimate or an overestimate? (b) Repeat part (a) using left endpoints.

a) Estimate the area under the graph of fsxd sx from x 0 to x 4 using four approximating rectangles and right endpoints. Sketch the graph and the rectangles. Is your estimate an underestimate or an overestimate? (b) Repeat part (a) using left endpoints.

(a) Estimate the area under the graph of fsxd 1 1 x 2 from x 21 to x 2 using three rectangles and right endpoints. Then improve your estimate by using six

a) Graph the function fsxd x 2 2 ln x 1 < x < 5 (b) Estimate the area under the graph of f using four approximating rectangles and taking the sample points to be (i) right endpoints and (ii) midpoints. In each case sketch the curve and the rectangles. (c) Improve your estimates in part (b) by using eight rectangles.

The speed of a runner increased steadily during the first three seconds of a race. Her speed at half-second intervals is given in the table. Find lower and upper estimates for the distance that she traveled during these three seconds. t ssd 0 0.5 1.0 1.5 2.0 2.5 3.0 v sftysd 0 6.2 10.8 14.9 18.1 19.4 20.2

Speedometer readings for a motorcycle at 12-second intervals are given in the table. (a) Estimate the distance traveled by the motorcycle during this time period using the velocities at the beginning of the time intervals. (b) Give another estimate using the velocities at the end of the time periods. (c) Are your estimates in parts (a) and (b) upper and lower estimates? Explain. t ssd 0 12 24 36 48 60 v sftysd 30 28 25 22 24 27

Measles pathogenesis The function fstd 2tst 2 21dst 1 1d can be used to model the measles pathogenesis curve in Figures 17 and 19. Suppose symptoms appear after 12 days. Use six subintervals and their midpoints to estimate the total amount of infection needed to develop symptoms.

Measles pathogenesis If a patient has had previous exposure to measles, the immune system responds more quickly. This results in a suppression of the level of virus ihe plasma during an infection. Suppose that such previous exposure causes the viral density in the plasma to be 3 5 of that in a patient with no previous exposure. This means that the level of virus in the plasma is given by 3 5 fstd, where fstd 2tst 2 21dst 1 1d as in Exercise 9. (a) Use subintervals of width 2 days and their midpoints to estimate the total amount of infection at t 12 days. (b) If 7848 cells per mL 3 days is the total amount of infection required to develop symptoms, use subintervals of width 2 days and their midpoints to estimate the day when symptoms will appear.

SARS incidence The table shows the number of people per day who died from SARS in Singapore at two-week intervals beginning on March 1, 2003. Date Deaths per day Date Deaths per day March 1 0.0079 April 26 0.5620 March 15 0.0638 May 10 0.4630 March 29 0.1944 May 24 0.2897 April 12 0.4435 Estimate the number of people who died of SARS in Singapore between March 1 and May 24, 2003, using both left endpoints and right endpoints. Source: Adapted from A. Gumel et al., Modelling Strategies for Controlling SARS Outbreaks, Proceedings of the Royal Society of London: Series B 271 (2004): 222332.

Niche overlap The extent to which species compete for resources is often measured by the niche overlap. If the horizontal axis represents a continuum of different resource types (for example, seed sizes for certain bird species), then a plot of the degree of preference for these resources is called a species niche. The degree of overlap of two species niches is then a measure of the extent to which they compete for resources. The niche overlap for a species is the fraction of the area under its preference curve that is also under the other species curve. The niches displayed in the figure are given by n1sxd sx 2 1ds3 2 xd 1 < x < 3 n2sxd sx 2 2ds4 2 xd 2 < x < 4 Estimate the niche overlap for species 1 using midpoints. (Choose the number of subintervals yourself.)

The velocity graph of a braking car is shown. Use it to estimate the distance traveled by the car while the brakes are applied.

The velocity graph of a car accelerating from rest to a speed of 120 kmyh over a period of 30 seconds is shown. Estimate the distance traveled during this period.

1517 Use Definition 2 to find an expression for the area under the graph of f as a limit. Do not evaluate the limit. 15. fsxd 2x x 2 1 1 , 1 < x < 3 16. fsxd x 2 1 s1 1 2x , 4 < x < 7 17. fsxd x cos x, 0 < x < y2

1517 Use Definition 2 to find an expression for the area under the graph of f as a limit. Do not evaluate the limit. 15. fsxd 2x x 2 1 1 , 1 < x < 3 16. fsxd x 2 1 s1 1 2x , 4 < x < 7 17. fsxd x cos x, 0 < x < y2

1517 Use Definition 2 to find an expression for the area under the graph of f as a limit. Do not evaluate the limit. 15. fsxd 2x x 2 1 1 , 1 < x < 3 16. fsxd x 2 1 s1 1 2x , 4 < x < 7 17. fsxd x cos x, 0 < x < y2

(a) Use Definition 2 to find an expression for the area under the curve y x 3 from 0 to 1 as a limit. (b) The following formula for the sum of the cubes of the first n integers is proved in Appendix E. Use it to evaluate the limit in part (a). 13 1 23 1 33 1 1 n3 F nsn 1 1d

Let A be the area under the graph of an increasing continuous function f from a to b, and let Ln and Rn be the approximations to A with n subintervals using left and right endpoints, respectively. (a) How are A, Ln, and Rn related? (b) Show that Rn 2 Ln b 2 a n f fsbd 2 fsadg

If A is the area under the curve y ex from 1 to 3, use Exercise 19 to find a value of n such that Rn 2 A , 0.0001.

(a) Express the area under the curve y x 5 from 0 to 2 as a limit. (b) Use a computer algebra system to find the sum in your expression from part (a). (c) Evaluate the limit in part (a).

Find the exact area of the region under the graph of y e2x from 0 to 2 by using a computer algebra system to evaluate the sum and then the limit in Example 3(a). Compare your answer with the estimate obtained in Example 3(b)

Find the exact area under the cosine curve y cos x from x 0 to x b, where 0 < b < y2. (Use a computer algebra system both to evaluate the sum and compute the limit.) In particular, what is the area if b y2?

(a) Let An be the area of a polygon with n equal sides inscribed in a circle with radius r. By dividing the polygon into n congruent triangles with central angle 2yn, show that An 1 2 nr 2 sinS 2 n D (b) Show that limnl` An r 2 . [Hint: Use Equation 2.4.6 on page 133.]