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All Textbook Solutions for Single Variable Calculus: Early Transcendentals, Volume I

25E 26E 27E 28E 29E 30E 31E Evaluate the integral. 14yyy2dy 33E 34E 35E 36E 37E 38E 39E 40E 41E 42E 43E 44E 45E 46E 47E 48E The area of the region that lies to the right of the y-axis and to the left of the parabola x = 2y y2 (the shaded region in the figure) is given by the integral 02(2yy2)dy. (Turn your head clockwise and think of the region as lying below the curve x = 2y y2 from y = 0 to y = 2.) Find the area of the region.The boundaries of the shaded region are the y-axis, the line y = 1, and the curve y=x4. Find the area of this region by writing x as a function of y and integrating with respect to y (as in Exercise 49).51E 52E If oil leaks from a tank at a rate of r(t) gallons per minute at time t, what does 0120r(t)dt represent?A honeybee population starts with 100 bees and increases at a rate of n(t) bees per week. What does 100+015n(t)dt represent?In Section 4.7 we defined the marginal revenue function R(x) as the derivative of the revenue function R(x), where x is the number of units sold. What does 10005000R(x)dx represent?If f(x) is the slope of a trail at a distance of x miles from the start of the trail, what does 35f(x)dx represent?57E 58E 59E 60E 61E The acceleration function (in m/s2) and the initial velocity are given for a particle moving along a line. Find (a) the velocity at time t and (b) the distance traveled during the given time interval. a(t) = 2t + 3, v(0) = 4, 0 t 3 63E 64E 65E 66E 67E 68E 69E 70E A bacteria population is 4000 at time t = 0 and its rate of growth is 1000 2t bacteria per hour after t hours. What is the population after one hour?72E Shown is the power consumption in the province of Ontario, Canada, for December 9, 2004 (P is measured in megawatts; t is measured in hours starting at midnight). Using the fact that power is the rate of change of energy, estimate the energy used on that day.1E 2E 3E 4E 5E 6E 7E 8E 9E 10E Evaluate the indefinite integral. cos(t/2)dt 12E 13E 14E 15E 16E 17E 18E 19E 20E 21E 22E 23E 24E 25E 26E 27E 28E Evaluate the indefinite integral. 5tsin(5t)dt 30E 31E 32E 33E 34E 35E 36E 37E 38E 39E 40E 41E 42E 43E 44E 45E 46E 47E 48E 49E 50E 51E 52E 53E 54E 55E 56E Evaluate the definite integral. 0/6sintcos2tdt 58E 59E 60E 61E 62E 63E 64E 65E 66E 67E 68E 69E 70E 71E 72E 73E 74E 75E 76E 77E Evaluate 01x1x4dx by making a substitution and interpreting the resulting integral in terms of an area.Which of the following areas are equal? Why?80E An oil storage tank ruptures at time t = 0 and oil leaks from the tank at a rate of r(t) = 100e0.01t liters per minute. How much oil leaks out during the first hour?82E 83E 84E Dialysis treatment removes urea and other waste products from a patients blood by diverting some of the bloodflow externally through a machine called a dialyzer. The rate at which urea is removed from the blood (in mg/min) is often well described by the equation u(t)=rVC0ert/V where r is the rate of flow of blood through the dialyzer (in mL/min), V is the volume of the patients blood (in mL), and C0 is the amount of urea in the blood (in mg) at time t = 0. Evaluate the integral 030u(t)dt and interpret it.86E 87E 88E If f is continuous on , prove that abf(x)dx=baf(x)dx For the case where f(x) 0 and 0 a b, draw a diagram to interpret this equation geometrically as an equality of areas.90E 91E 92E 93E 94E (a) Write an expression for a Riemann sum of a function f. Explain the meaning of the notation that you use. (b) If f(x) 0, what is the geometric interpretation of a Riemann sum? Illustrate with a diagram. (c) If f(x) takes on both positive and negative values, what is the geometric interpretation of a Riemann sum? Illustrate with a diagram.2RCC 3RCC 4RCC 5RCC Suppose a particle moves back and forth along a straight line with velocity v(t), measured in feet per second, and acceleration a(t). (a) What is the meaning of 60120v(t)dt? (b) What is the meaning of 60120v(t)dt? (c) What is the meaning of 60120a(t)dt?7RCC 8RCC 9RCC 1RQ 2RQ 3RQ 4RQ 5RQ 6RQ 7RQ 8RQ 9RQ 10RQ 11RQ 12RQ 13RQ 14RQ 15RQ 16RQ 17RQ 18RQ Use the given graph of f to find the Riemann sum with six subintervals. Take the sample points to be (a) left endpoints and (b) midpoints. In each case draw a diagram and explain what the Riemann sum represents.2RE 3RE 4RE 5RE 6RE 7RE Evaluate: (a) 01ddx(earctanx)dx (b) ddx01(earctanx)dx (c) ddx0x(earctant)dt 9RE 10RE 11RE 12RE 13RE 14RE 15RE 16RE 17RE 18RE 19RE 20RE 21RE 22RE 23RE 24RE 25RE 26RE 27RE 28RE 29RE 30RE 31RE 32RE 33RE 34RE 35RE 36RE 37RE 38RE 39RE 40RE 41RE 42RE 43RE 44RE 45RE 46RE 47RE 48RE 49RE 50RE 51RE Use Property 8 of integrals to estimate the value of the integral. 351x+1dx 53RE 54RE 55RE Use the properties of integrals to verify the inequality. 01xsin1xdx/4 Use the Midpoint Rule with n = 6 to approximate 03sin(x3)dx.58RE 59RE A radar gun was used to record the speed of a runner at the times given in the table. Use the Midpoint Rule to estimate the distance the runner covered during those 5 seconds. t(s) v(m/s) 0 0 0.5 4.67 1.0 7.34 1.5 8.86 2.0 9.73 2.5 10.22 3.0 10.51 3.5 10.67 4.0 10.76 4.5 10.81 5.0 10.81 61RE 62RE 63RE 66RE 69RE 70RE 71RE 72RE 1P 2P 3P 5P 6P 7P The figure shows two regions in the first quadrant: A(t) is the area under the curve y = sin(x2) from 0 to t, and B(t) is the area of the triangle with vertices O, P, and (t, 0). Find limt0+[A(t)/B(t)]. FIGURE FOR PROBLEM 8 9P 10P 11P 14P 15P 18P 19P Find the area of the shaded region.2E 3E 4E 5E 6E Sketch the region enclosed by the given curves. Decide whether to integrate with respect to x or y. Draw a typical approximating rectangle and label its height and width. Then find the area of the region. y = (x 2)2, y = x Sketch the region enclosed by the given curves. Decide whether to integrate with respect to x or y. Draw a typical approximating rectangle and label its height and width. Then find the area of the region. y = x2 4x, y = 2x Sketch the region enclosed by the given curves. Decide whether to integrate with respect to x or y. Draw a typical approximating rectangle and label its height and width. Then find the area of the region. y = 1/x, y = 1/x2, x = 2 Sketch the region enclosed by the given curves. Decide whether to integrate with respect to x or y. Draw a typical approximating rectangle and label its height and width. Then find the area of the region. y = sin x, y =2x/, x 0 11E 12E 13E 14E 15E 16E 17E 18E 19E Sketch the region enclosed by the given curves and find its area. x = y4, y=2x, y = 0 21E Sketch the region enclosed by the given curves and find its area. y = x3, y = x 23E 24E 25E Sketch the region enclosed by the given curves and find its area. y = sinh x, y = ex, x = 0, x = 2 Sketch the region enclosed by the given curves and find its area. y = 1/x, y = x, y=14x, .x 0 28E 29E 30E 31E 32E Use calculus to find the area of the triangle with the given vertices. (0, 0), (3, 1), (1, 2)Use calculus to find the area of the triangle with the given vertices. (2, 0), (0, 2), (1, 1)35E Evaluate the integral and interpret it as the area of a region. Sketch the region. 113x2xdx 37E 38E 39E 40E 41E 42E 43E 44E Sketch the region in the xy-plane defined by the inequalities x 2y2 0, 1 x |y| 0 and find its area.47E The widths (in meters) of a kidney-shaped swimming pool were measured at 2-meter intervals as indicated in the figure. Use the Midpoint Rule to estimate the area of the pool.A cross-section of an airplane wing is shown. Measurements of the thickness of the wing, in centimeters, at 20-centimeter intervals are 5.8, 20.3, 26.7, 29.0, 27.6, 27.3, 23.8, 20.5, 15.1, 8.7, and 2.8. Use the Midpoint Rule to estimate the area of the wings cross-section.

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