Problem 1
(a) By reading values from the given graph of $ f $, use five rectangles to find a lower estimate and an upper estimate for the area under the given graph of $ f $ from $ x = 0 $ to $ x = 10 $. In each case sketch the rectangles that you use.
(b) Find new estimates using ten rectangles in each case.
Problem 2
(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) $ L_{6} $ (sample points are left endpoints)
(ii) $ R_{6} $ (sample points are right endpoints)
(iii) $ M_{6} $ (sample points are midpoints)
(b) Is $ L_{6} $ an underestimate or overestimate of the true area?
(c) Is $ R_{6} $ an underestimate or overestimate of the true area?
(d) Which of the numbers $ L_{6} $, $ R_{6} $, or $ M_{6} $ gives the best estimate? Explain.
Problem 3
(a) Estimate the area under the graph of $ f(x) = 1/x $ from $ x = 1 $ to $ x =2 $ 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.
Problem 4
(a) Estimate the area under the graph of $ f(x) = \sin x $ from $ x = 0 $ to $ x = \pi/2 $ 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.
Problem 5
(a) Estimate the area under the graph of $ f(x) = 1 + x^2 $ from $ x = -1 $ to $ x = 2 $ using three rectangles and right endpoints. Then improve your estimate by using six rectangles. Sketch the curve and the approximating rectangles.
(b) Repeat part (a) using left endpoints.
(c) Repeat part (a) using midpoints.
(d) From your sketches in parts (a)-(c), which appears to be the best estimate?
Problem 6
(a) Graph the function $$ f(x) = x - 2 \ln x \hspace{10mm} 1 \le x \le 5 $$
(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.
Problem 7
Evaluate the upper and lower sums for $ f(x) = 2 + \sin x $, $ 0 \le x \le \pi $, with $ n $ = 2, 4, and 8. Illustrate with diagrams like Figure 14.
Problem 8
Evaluate the upper and lower sums for $ f(x) = 1 + x^2 $, $ -1 \le x \le 1 $, with $ n $ = 3 and 4. Illustrate with diagrams like Figure 14.
Problem 9
With a programmable calculator (or a computer), it is possible to evaluate the expressions for the sums of areas of approximating rectangles, even for large values of $ n $, using looping. (On a TI use the Is> command or a For-EndFor loop, on a Casio use Isz, on an HP or in BASIC use a FOR-NEXT loop.) Compute the sum of the areas of approximating rectangles using equal subintervals and right endpoints for $ n $ = 10, 30, 50, and 100. Then guess the value of the exact area.
The region under $ y = x^4 $ from 0 to 1
Problem 10
With a programmable calculator (or a computer), it is possible to evaluate the expressions for the sums of areas of approximating rectangles, even for large values of $ n $, using looping. (On a TI use the Is> command or a For-EndFor loop, on a Casio use Isz, on an HP or in BASIC use a FOR-NEXT loop.) Compute the sum of the areas of approximating rectangles using equal subintervals and right endpoints for $ n $ = 10, 30, 50, and 100. Then guess the value of the exact area.
The region under $ y = \cos x $ from 0 to $ \pi/2 $
Problem 11
Some computer algebra systems have commands that will draw approximating rectangles and evaluate the sums of their areas, at least if $ x_{i}^{*} $ is a left or right endpoint. (For instance, in Maple use leftbox, rightbox, leftsum, and rightsum.)
(a) If $ f(x) = 1/(x^2 + 1) $, $ 0 \le x \le 1 $, find the left and right sums for $ n $ = 10, 30, and 50.
(b) Illustrate by graphing the rectangles in part (a).
(c) Show that the exact area under $ f $ lies between 0.780 and 0.791.
Problem 12
(a) If $ f(x) = \ln x $, $ 1 \le x \le 4 $, use the commands discussed in Exercise 11 to find the left and right sums for $ n $ = 10, 30, and 50.
(b) Illustrate by graphing the rectangles in part (a).
(c) Show that the exact area under $ f $ lies between 2.50 and 2.59.
Problem 13
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.
Problem 14
The table shows a speedometer readings at 10-second intervals during a 1-minute period for a car racing at the Daytona International Speedway in Florida.
(a) Estimate the distance the race car traveled 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.
Problem 15
Oil leaked from a tank at a rate of $ r(t) $ liters per hour. The rate decreased as time passed and values of the rate at two-hour time intervals are shown in the table. Find lower and upper estimates for the total amount of oil that leaked out.
Problem 16
When we estimate distances from velocity data, it is sometimes necessary to use times $ t_0, t_1, t_2, t_3, ... $ that are not equally spaced. We can still estimate distances using the time periods
$ \Delta t_i = t_i - t_{i-1} $. For example, on May 7, 1992, the space shuttle $ Endeavour $ was launched on mission STS-49, the purpose of which was to install a new perigee kick motor in an Intelsat communications satellite. The table, provided by NASA, gives the velocity data for the shuttle between liftoff and the jettisoning of the solid rocket boosters. Use these data to estimate the height above the earth's surface of the $ Endeavour $, 62 seconds after liftoff.
Problem 17
The velocity graph of a braking car is shown. Use it to estimate the distance traveled by the car while the brakes are applied.
Problem 18
The velocity graph of a car accelerating from rest to a speed of 120 km/h over a period of 30 seconds is shown. Estimate the distance traveled during this period.
Problem 19
In someone affected with measles, the virus level $ N $ (measured in number of infected cells per mL of blood plasma) reaches a peak density at about $ t = 12 $ days (when a rash appears) and then decreases fairly rapidly as a result of immune response. The area under the graph of $ N(t) $ from
$ t = 0 $ to $ t = 12 $ (as shown in the figure) is equal to the total amount of infection needed to develop symptoms (measured in density of infected cells x time). The function $ N $ has been modeled by the function $$ f(t) = -t(t - 21)(t + 1) $$
Use this model with six subintervals and their midpoints to estimate the total amount of infection needed to develop symptoms of measles.
Problem 20
The table shows the number of people per day who died from SARS in Singapore at two-week intervals beginning on March 1, 2003.
(a) By using an argument similar to that in Example 4, 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.
(b) How would you interpret the number of SARS deaths as an area under a curve?
Problem 21
Use Definition 2 to find an expression for the area under the graph of $ f $ as a limit. Do not evaluate the limit.
$ f(x) = \dfrac{2x}{x^2 + 1}, \hspace{5mm} 1 \le x \le 3 $
Problem 22
Use Definition 2 to find an expression for the area under the graph of $ f $ as a limit. Do not evaluate the limit.
$ f(x) = x^2 + \sqrt{1 + 2x}, \hspace{5mm} 4 \le x \le 7 $
Problem 23
Use Definition 2 to find an expression for the area under the graph of $ f $ as a limit. Do not evaluate the limit.
$ f(x) = \sqrt{\sin x}, \hspace{5mm} 0 \le x \le \pi $
Problem 24
Determine a region whose area is equal to the given limit. Do not evaluate the limit.
$ \displaystyle \lim_{n \to \infty} \sum_{i = 1}^{n} \frac{3}{n} \sqrt{1 +\frac{3i}{n}} $
Problem 25
Determine a region whose area is equal to the given limit. Do not evaluate the limit.
$ \displaystyle \lim_{n \to \infty} \sum_{i = 1}^{n} \frac{\pi}{4n} \tan{\frac{i \pi}{4n}} $
Problem 26
(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).
$$ 1^3 + 2^3 + 3^3 + \cdots + n^3 = \biggl[ \frac{n(n + 1)}{2} \biggr]^2 $$
Problem 27
Let $ A $ be the area under the graph of an increasing continuous function $ f $ from $ a $ to $ b $, and let $ L_n $ and $ R_n $ be the approximations to $ A $ with $ n $ subintervals using left and right endpoints, respectively.
(a) How are $ A $, $ L_n $, and $ R_n $ related?
(b) Show that $$ R_n - L_n = \frac{b - a}{n} [f(b) - f(a)] $$
Then draw a diagram to illustrate this equation by showing that the $ n $ rectangles representing $ R_n - L_n $ can be reassembled to form a single rectangle whose area is the right side of the equation.
(c) Deduce that $$ R_n - A < \frac{b - a}{n} [f(b) - f(a)] $$
Problem 28
If $ A $ is the area under the curve $ y = e^x $ from 1 to 3, use Exercise 27 to find a value of $ n $ such that $ R_n - A < 0.0001 $.
Problem 29
(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).
Problem 30
Find the exact area of the region under the graph of $ y = e^{-x} $ 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).
Problem 31
Find the exact area under the cosine curve $ y = \cos x $ from $ x = 0 $ to $ x = b $, where $ 0 \le b \le \pi/2 $. (Use a computer algebra system both to evaluate the sum and compute the limit.) In particular, what is the area if $ b = \pi/2 $?
Problem 32
(a) Let $ A_n $ 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 $ 2\pi/n $, show that
$$ A_n = \frac{1}{2} nr^2 \sin \biggl( \frac{2 \pi}{n} \biggr) $$
(b) Show that $ \displaystyle \lim_{n \to \infty} A_n = \pi r^2 $. [$ Hint: $ Use Equation 3.3.2 on page 191.]
