Find the area of the region between the curves. y = x2 + 1 and y = -x2 - 1 from x = -1 to x = 1
> Assuming that a country’s population is now 3 million and is growing exponentially with growth constant .02, what will be the average population during the next 50 years?
> During a certain 12-hour period, the temperature at time t (measured in hours from the start of the period) was T(t) = 47 + 4t – 1/3 t2 degrees. What was the average temperature during that period?
> Determine the average value of f (x) over the interval from x = a to x = b, where f (x) = 1/√x; a = 1, b = 9.
> Determine the average value of f (x) over the interval from x = a to x = b, where f (x) = 1/x; a = 1/3, b = 3.
> Determine the average value of f (x) over the interval from x = a to x = b, where f (x) = 2; a = 0, b = 1.
> Shade the portion of Fig. 23 whose area is given by the integral 0∫2 [ f (x) - g (x)] dx + 2∫4 [h(x) - g (x)] dx. Figure 23: 3 1 y = h(x) y = g(x) 2 3 y = f(x) 4 5
> Write a definite integral or sum of definite integrals that gives the area of the shaded portions in Fig. 22. Figure 22: 2 1 U 0 -1 -2- 1 y = f(x) 不 3 4 y = g(r) M
> Write a definite integral or sum of definite integrals that gives the area of the shaded portions in Fig. 21. Figure 21: -1 0 y y = f(x) 2 3 4
> Use a graphing utility to find the intersection points of the curves, and then use the utility to find the area of the region bounded by the curves. y = 1/x, y = 3 - x
> Use a graphing utility to find the intersection points of the curves, and then use the utility to find the area of the region bounded by the curves. y = √(x + 1), y = (x - 1)2
> Determine the following: ∫e dx
> Use a graphing utility to find the intersection points of the curves, and then use the utility to find the area of the region bounded by the curves. y = 5 - (x - 2)2, y = ex
> Use a graphing utility to find the intersection points of the curves, and then use the utility to find the area of the region bounded by the curves. y = ex, y = 4x + 1
> The velocity of an object moving along a line is given by υ(t) = t2 + t - 2 feet per second. (a) Find the displacement of the object as t varies in the interval 0 ≤ t ≤ 3. Interpret this displacement using area under the graph of υ(t). (b) Find the total
> The velocity of an object moving along a line is given by υ(t) = 2t2 - 3t + 1 feet per second. (a) Find the displacement of the object as t varies in the interval 0 ≤ t ≤ 3. (b) Find the total distance traveled by the object during the interval of time 0
> Cars A and B start at the same place and travel in the same direction, with velocities after t hours given by the functions Ï…A(t) and Ï…B(t) in Fig. 29. (a) What does the area between the two curves from t = 0 to t = 1 represent? (b)
> Two rockets are fired simultaneously straight up into the air. Their velocities (in meters per second) are υ1(t) and υ2(t), and υ1(t) ≥ υ2(t) for t ≥ 0. Let A denote the area of the region between the graphs of y = υ1(t) and y = υ2(t) for 0 ≤ t ≤ 10. Wha
> The marginal profit for a certain company is MP1(x) = -x2 + 14x - 24. The company expects the daily production level to rise from x = 6 to x = 8 units. The management is considering a plan that would have the effect of changing the marginal profit to M2(
> After an advertising campaign, a company’s marginal profit was adjusted up from M1(x) = 2x2 - 3x + 11, before advertising, to M2(x) = 2x2 - 2.4x + 8, after advertising. Here x denotes the number of units produced, and M1(x) and M2(x) are measured in thou
> Refer to Exercise 39. The rate of new tree growth (in millions of cubic meters per year) in the Sudan t years after 1980 is given approximately by the function g(t) = 50 - 6.03e0.09t. Set up the definite integral giving the amount of depletion of the for
> Deforestation is one of the major problems facing sub-Saharan Africa. Although the clearing of land for farming has been the major cause, the steadily increasing demand for fuel wood has also become a significant factor. Figure 28 summarizes projections
> Determine the following: ∫ (7/2x3 – 3√x) dx
> Find all antiderivatives of each following function: f (x) = 9x8
> Suppose that the velocity of a car at time t is υ(t) = 40 + 8/(t + 1)2 kilometers per hour. (a) Compute the area under the velocity curve from t = 1 to t = 9. (b) What does the area in part (a) represent?
> Some food is placed in a freezer. After t hours, the temperature of the food is dropping at the rate of r(t) degrees Fahrenheit per hour, where r(t) = 12 + 4/(t + 3)2. (a) Compute the area under the graph of y = r(t) over the interval 0 ≤ t ≤ 2. (b) What
> Let M(x) be a company’s marginal cost at production level x. Give an economic interpretation of the number 0∫100 M(x)dx.
> Let M(x) be a company’s marginal profit at production level x. Give an economic interpretation of the number 44∫48 M(x)dx.
> Suppose that the marginal profit function for a company is P (x) = 100 + 50x - 3x2 at production level x. (a) Find the extra profit earned from the sale of 3 additional units if 5 units are currently being produced. (b) Describe the answer to part (a) as
> Suppose that the marginal cost function of a handbag manufacturer is C (x) = 3/32 x2 - x + 200 dollars per unit at production level x (where x is measured in units of 100 handbags). (a) Find the total cost of producing 6 additional units if 2 units are c
> After t hours of operation, an assembly line is producing lawn mowers at the rate of r(t) = 21 – 4/5 t mowers per hour. (a) How many mowers are produced during the time from t = 2 to t = 5 hours? (b) Represent the answer to part (a) as an area.
> A helicopter is rising straight up in the air. Its velocity at time t is υ(t) = 2t + 1 feet per second. (a) How high does the helicopter rise during the first 5 seconds? (b) Represent the answer to part (a) as an area.
> Find the area of the region bounded by y = 1/x, y = 4x, and y = x/2, for x ≥ 0. (The region resembles the shaded region in Exercise 29.) Figure 27: Y /y = 8x y y=x x
> Find the area in Fig. 27 of the region bounded by y = 1/x2, y = x, and y = 8x, for x ≥ 0. Figure 27: Y /y = 8x y y=x x
> Determine the following: ∫3e-2x dx
> Find the area of the region between y = x2 and y = 1/x2 (a) from x = 1 to x = 4, (b) from x = 1/2 to x = 4.
> Find the area of the region between y = x2 - 3x and the x-axis (a) from x = 0 to x = 3, (b) from x = 0 to x = 4, (c) from x = -2 to x = 3.
> Find the area of the region bounded by the curves. y = 4/x and y = 5 - x
> Find the area of the region bounded by the curves. y = 8x2 and y = √x
> Find the area of the region bounded by the curves. y = x3 and y = 2x2
> Find the area of the region bounded by the curves. y = x(x2 - 1) and the x-axis
> Find the area of the region bounded by the curves. y = x2 - 1 and y = 3
> Find the area of the region bounded by the curves. y = -x2 + 6x - 5 and y = 2x - 5
> Find the area of the region bounded by the curves. y = 4x(1 - x) and y = 3/4
> Find the area of the region bounded by the curves. y = x2 and y = x
> Determine the following: ∫ (7/2x3 – 3√x) dx
> Find the area of the region between the curves. y = e2x and y = 1 - x from x = 0 to x = 1
> Find the area of the region between the curves. y = ex and y = 1/x2 from x = 1 to x = 2
> Find the area of the region between the curves. y = x(2 - x) and y = 2 from x = 0 to x = 2
> Find the area of the region between the curves. y = x2 - 6x + 12 and y = 1 from x = 0 to x = 4
> Find the area of the region between the curves. y = 2x2 and y = 8 (a horizontal line) from x = -2 to x = 2
> Find the area of the region between the curve and the x-axis. f (x) = e-x + 2 from -1 to 2
> Find the area of the region between the curve and the x-axis. f (x) = ex - 3 from 0 to ln 3
> Find the area of the region between the curve and the x-axis. f (x) = x2 + 6x + 5 from 0 to 1.
> Find the area of the region between the curve and the x-axis. f (x) = x2 - 2x - 3, from 0 to 2.
> Determine the following: ∫ (x - 2x2 + 1/3x) dx
> Find the area of the region between the curve and the x-axis. f (x) = x(x2 - 1), from -1 to 1.
> Find the area of the region between the curve and the x-axis. f (x) = 1 - x2, from -2 to 2
> Let g(x) be the function pictured in Fig. 26. Determine whether 1 > 0 g(x)dx is positive, negative, or zero. Figure 26: 2 1 -2 -3 y y = g(x) +x + 1 2 3 4 5/6 7 8
> Let f (x) be the function pictured in Fig. 25. Determine whether 1 > 0 f (x)dx is positive, negative, or zero. Figure 25: 3 2 1 -1 Y 12 y = f(x) 3 4 5 7 8 8
> Shade the portion of Fig. 24 whose area is given by the integral 0∫1L [ f (x) - g (x)] + 1∫2 [ g (x) - f (x)] dx. Figure 24: 2 y = f(x) 1 y = g(x) 0 У х а 1 2
> Compute the area of the shaded region in two different ways: (a) by using simple geometric formulas; (b) by applying Theorem I. Y f(x) = -x 2- -2 2 0
> Compute the area of the shaded region in two different ways: (a) by using simple geometric formulas; (b) by applying Theorem I. y 1 0 1 (x)=1 1 2 x
> Compute the area of the shaded region in two different ways: (a) by using simple geometric formulas; (b) by applying Theorem I. 2 Y 0 12 f(x) = 2 23 3 4
> Evaluate a Riemann sum to approximate the area under the graph of f (x) on the given interval, with points selected as specified. f (x) = x√(1 + x2); 1 ≤ x ≤ 3, n = 20, midpoints of subintervals f
> Evaluate a Riemann sum to approximate the area under the graph of f (x) on the given interval, with points selected as specified. f (x) = x√(1 + x2); 1 ≤ x ≤ 3, n = 20, midpoints of subintervals
> Determine the following: ∫ (2/√x + 2√x) dx
> Compute the area under the graph of y = 1 / (1 + x2) from 0 to 5.
> The area under the graph of the function e-x2 plays an important role in probability. Compute this area from -1 to 1.
> We show that, as the number of subintervals increases indefinitely, the Riemann sum approximation of the area under the graph of f (x) = x2 from 0 to 1 approaches the value 1/3 , which is the exact value of the area. Partition the interval [0, 1] into n
> We show that, as the number of subintervals increases indefinitely, the Riemann sum approximation of the area under the graph of f (x) = x2 from 0 to 1 approaches the value 1/3 , which is the exact value of the area. Verify the given formula for n = 1, 2
> A farmer wants to divide the lot in Fig. 18 into two lots of equal area by erecting a fence that extends from the road to the river as shown. Determine the location of the fence. Figure 18: 10 20 30 40 Road 50 60 70 80 06 ft 40 ft 35 ft 30 ft 25 ft
> Estimate the area (in square feet) of the residential lot in Fig. 17. Figure 17: 0 20 60 100 140 160 106 ft 101 ft 100 ft 113 ft
> Use a Riemann sum with n = 5 and midpoints to estimate the area under the graph of f (x) = √(1 - x2) on the interval 0 ≤ x ≤ 1. The graph is a quarter circle, and the area under the graph is .7854
> The graph of the function f (x) = √(1 - x2) on the interval -1 ≤ x ≤ 1 is a semicircle. The area under the graph is 1/2 π(1)2 = π/2 = 1.57080, to five decimal places. Use a R
> Use a Riemann sum with n = 4 and right endpoints to estimate the area under the graph of f (x) = 2x - 4 on the interval 2 ≤ x ≤ 3. Then, repeat with n = 4 and midpoints. Compare the answers with the exact answer, 1
> Use a Riemann sum with n = 4 and left endpoints to estimate the area under the graph of f (x) = 4 - x on the interval 1 ≤ x ≤ 4. Then repeat with n = 4 and midpoints. Compare the answers with the exact answer, 4.5,
> Determine the following: ∫x√x dx
> Use a Riemann sum to approximate the area under the graph of f (x) in Fig. 14 on the given interval, with selected points as specified. Draw the approximating rectangles. 1 ≤ x ≤ 7, n = 3, midpoints of subintervals
> Use a Riemann sum to approximate the area under the graph of f (x) in Fig. 14 on the given interval, with selected points as specified. Draw the approximating rectangles. 4 ≤ x ≤ 9, n = 5, right endpoints Riemann
> Use a Riemann sum to approximate the area under the graph of f (x) in Fig. 14 on the given interval, with selected points as specified. Draw the approximating rectangles. 3 ≤ x ≤ 7, n = 4, left endpoints Riemann s
> Use a Riemann sum to approximate the area under the graph of f (x) in Fig. 14 on the given interval, with selected points as specified. Draw the approximating rectangles. 0 ≤ x ≤ 8, n = 4, midpoints of subintervals
> Use a Riemann sum to approximate the area under the graph of f (x) on the given interval, with selected points as specified. f (x) = ln x; 2 ≤ x ≤ 4, n = 5, left endpoints Riemann sum: Ricmann sum. f(x1) Δx +
> Use a Riemann sum to approximate the area under the graph of f (x) on the given interval, with selected points as specified. f (x) = e-x; 2 ≤ x ≤ 3, n = 5, right endpoints Riemann sum: Ricmann sum. f(x1) Δx +
> Use a Riemann sum to approximate the area under the graph of f (x) on the given interval, with selected points as specified. f (x) = x3; 0 ≤ x ≤ 1, n = 5, right endpoints Riemann sum: Ricmann sum. f(x1) Δx +
> Use a Riemann sum to approximate the area under the graph of f (x) on the given interval, with selected points as specified. f (x) = x3; 1 ≤ x ≤ 3, n = 5, left endpoints Riemann sum: Ricmann sum. f(x1) Δx + f
> Use a Riemann sum to approximate the area under the graph of f (x) on the given interval, with selected points as specified. f (x) = x2; -2 ≤ x ≤ 2, n = 4, midpoints of subintervals Riemann sum: Ricmann sum.
> Use a Riemann sum to approximate the area under the graph of f (x) on the given interval, with selected points as specified. f (x) = x2; 1 ≤ x ≤ 3, n = 4, midpoints of subintervals Riemann sum: Ricmann sum. f
> Determine the following: ∫1/7x dx
> Determine Δx and the midpoints of the subintervals formed by partitioning the given interval into n subintervals. 3 ≤ x ≤ 5; n = 5
> Determine Δx and the midpoints of the subintervals formed by partitioning the given interval into n subintervals. 1 ≤ x ≤ 4; n = 5
> Determine Δx and the midpoints of the subintervals formed by partitioning the given interval into n subintervals. 0 ≤ x ≤ 3; n = 6
> Determine Δx and the midpoints of the subintervals formed by partitioning the given interval into n subintervals. 0 ≤ x ≤ 2; n = 4
> Find the real number b 7 0 so that the area under the graph of y = x2 from 0 to b is equal to the area under the graph of y = x3 from 0 to b.
> Find the real number b > 0 so that the area under the graph of y = x3 from 0 to b is equal to 4.
> Find the area under each of the given curves. y = e3x; x = - 1/3 to x = 0
> Find the area under each of the given curves. y = (x - 3)4; x = 1 to x = 4
> Find the area under each of the given curves. y = √x; x = 0 to x = 4
> Find the area under each of the given curves. y = 3x2 + x + 2ex/2; x = 0 to x = 1
> Determine the following: ∫ (2/x + x/2) dx
> Find the area under each of the given curves. y = 3x2; x = -1 to x = 1
