(a). If f is continuous, prove that fπ/20 f (cos x) dx = fπ/20 f (sin x) dx (b). Use part (a) to evaluate fπ/20 cos2 x dx and x fπ/20 sin 2x dx.
> Determine whether the sequence converges or diverges. If it converges, find the limit. sin 2n 1 + + Jn
> Determine whether the series is convergent or divergent. If it is convergent, find its sum. 1 Σ п(n + 1) A-1 +
> Find the radius of convergence and interval of convergence of the series. (4x + 1)" n? 00 n-1
> Determine whether the series is convergent or divergent. If it is convergent, find its sum. E [(0.8)*-1 – (0.3)*]
> Determine whether the series is convergent or divergent. If it is convergent, find its sum. Σ 2 arctan n
> Determine whether the series is convergent or divergent. If it is convergent, find its sum. -1
> Determine whether the series is convergent or divergent. If it is convergent, find its sum. 1+ 3" 24
> Determine whether the series is convergent or divergent. If it is convergent, find its sum. 1+ 24 Σ 34
> Determine whether the series is convergent or divergent. If it is convergent, find its sum. 00 Σ cos A-1
> Determine whether the series is convergent or divergent. If it is convergent, find its sum. k? Σ k? – 1 00 - 1 k-2
> Determine whether the series is convergent or divergent. If it is convergent, find its sum. k(k + 2) Σ -i (k + 3)2 k-1
> Explain what it means to say that ∑∞n-1 an = 5.
> Determine whether the series is convergent or divergent. If it is convergent, find its sum. n - 1 Σ A-1 3n Зп — 1
> Show that the series is convergent. How many terms of the series do we need to add in order to find the sum to the indicated accuracy? E (-1)*-'ne- (1 error|< 0.01)
> Determine whether the geometric series is convergent or divergent. If it is convergent, find its sum. Σ n-0
> Determine whether the geometric series is convergent or divergent. If it is convergent, find its sum. 00 Σ A-o 3*+1
> Determine whether the geometric series is convergent or divergent. If it is convergent, find its sum. 10" Σ (-9)*-1
> Determine whether the geometric series is convergent or divergent. If it is convergent, find its sum. 00 E 6(0.9)“-! A-1 n-1
> Determine whether the geometric series is convergent or divergent. If it is convergent, find its sum. 1 + 0.4 + 0.16 + 0.064 +
> Determine whether the geometric series is convergent or divergent. If it is convergent, find its sum. 10 – 2 + 0.4 – 0.08 + ·
> Determine whether the geometric series is convergent or divergent. If it is convergent, find its sum. 4 + 3 ++ + ...
> Determine whether the geometric series is convergent or divergent. If it is convergent, find its sum. 3 – 4 +4 - + ... 16
> (a). Explain the difference between (b). Explain the difference between aj and 内I and E aj
> (a). What is the difference between a sequence and a series? (b). What is a convergent series? What is a divergent series?
> Show that the series is convergent. How many terms of the series do we need to add in order to find the sum to the indicated accuracy? (-1)*+1 (lerror|< 0.00005) A-1
> (a). What is an alternating series? (b). Under what conditions does an alternating series converge? (c). If these conditions are satisfied, what can you say about the remainder after n terms?
> On what interval is the curve y = fx0 t2/t2 + t + 2, dt concave downward?
> If f (x) = fsinxo√1 + t2 dt and g (y) = fy3 f(x) dx, find g"(π/6).
> If f (x) = fx0 (1 – t2) et2 dt, on what interval is f increasing?
> Evaluate the integral. f01 10x dx
> Evaluate the integral. f12 x-2 dx
> Evaluate the integral. f1/2√3/2 6/√1 – t2 dt
> find f. f'(x) = √x (6 + 5x) f (1) = 10
> find f. f'(x) = 8x3 + 12x + 3, f (1) = 6
> Find f. f"(x) = 1 - 6x, f (0) = 8
> Find f. f"(x) = 6x + sin x
> Find f. f"(x) = 6x + 12x2
> Use Part 1 of the Fundamental Theorem of Calculus to find the derivative of the function. *cos X y = * (1 + v?)1º dv Jsin x
> Find the antiderivative F of f that satisfies the given condition. Check your answer by comparing the graphs of f and F. f(x) — 4 — 3(1 + х?)-1, F(1) = 0
> Find the antiderivative F of f that satisfies the given condition. Check your answer by comparing the graphs of f and F. f(x) = 5x* – 2x', F(0)=4
> Find the most general antiderivative of the function. (Check your answer by differentiation.) f(x) = 27 + 6 cos x COS %3D
> Find the most general antiderivative of the function. (Check your answer by differentiation.) g(8) = cos e – 5 sin e
> Find the most general antiderivative of the function. (Check your answer by differentiation.) f(x) = 3e* + 7 sec?x
> Let S be the solid obtained by rotating the region shown in the figure about the y-axis. Sketch a typical cylindrical shell and find its circumference and height. Use shells to find the volume of S. Do you think this method is preferable to slicing? Expl
> Evaluate the limit by first recognizing the sum as a Riemann sum for a function defined on [0, 1]. 2 3 lim +
> Evaluate the limit by first recognizing the sum as a Riemann sum for a function defined on [0, 1]. lim 4
> The area labeled B is three times the area labeled A. Express a in terms of b. y. y= e" y= e" B A a
> Suppose h is a function such that h (1) = -2, h'(1) = 2, h"(1) = 3, h (2) = 6, h'(2) = 5, h"(2) = 13, and h" is continuous everywhere. Evaluate f21 h"(u) du.
> A manufacturer of corrugated metal roofing wants to produce panels that are 28 in. wide and 2 in. thick by processing flat sheets of metal as shown in the figure. The profile of the roofing takes the shape of a sine wave. Verify that the sine curve has e
> If a and b are positive numbers, show that r(1 – x)* dx = [ x*(1 – x)* dx
> If f is continuous on R, prove that fbaf (x + c) dx = fb+ca+c f (x) dx For the case where f (x) > 0, draw a diagram to interpret this equation geometrically as an equality of areas.
> If f is continuous on R, prove that fbaf (-x) dx = f-1-bf (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.
> If f is continuous and f90 f (x) dx = 4, find f30 xf (x2) dx.
> If f is continuous and f40f (x) dx = 10, find f20f (2x) dx.
> The velocity graph of a car accelerating from rest to a speed of 120 km/hover a period of 30 seconds is shown. Estimate the distance traveled during this period. (km/h) 80 40 10 20 30 (seconds)
> Alabama Instruments Company has set up a production line to manufacture a new calculator. The rate of production of these calculators after weeks is (Notice that production approaches 5000 per week as time goes on, but the initial production is lower b
> Breathing is cyclic and a full respiratory cycle from the beginning of inhalation to the end of exhalation takes about 5 s. The maximum rate of air flow into the lungs is about 0.5 L/s. This explains, in part, why the function f (t) = ½ sin (2πt/5) has o
> Find the value of the constant C for which the integral converges. Evaluate the integral for this value of C. C 3r + 1) dx x² + 1
> Use Part 1 of the Fundamental Theorem of Calculus to find the derivative of the function. 3r u? - 1 du J2x u? + g(x) = Hint: "f(u) du = f(u) du + f* f(u) du /2x
> Find the value of the constant C for which the integral converges. Evaluate the integral for this value of C. 1 C x² + 4 dx X + 2
> Show that f∞0e-x2 dx = f10 √-ln y, dy by interpreting the integrals as areas.
> Show that fπ0x2e-x2 dx = 1/2 f∞0e-x2 dx.
> Estimate the numerical value of f∞0e-x2 dx by writing it as the sum of f40e-x2 dx and f∞4e-x2 dx. Approximate the first integral by using Simpson’s Rule with n = 8 and show that the second integral is smaller than f∞4e-4x dx, which is less than 0.0000001
> Determine how large the number a has to be so that 1 dx < 0.001 x2 + 1
> Astronomers use a technique called stellar stereography to determine the density of stars in a star cluster from the observed (two-dimensional) density that can be analyzed from a photograph. Suppose that in a spherical cluster of radius R the density of
> As we will see in Section 7.4, a radioactive substance decays exponentially: The mass at time t is m (t) = m (0) ekt, where m (0) is the initial mass and is a negative constant. The mean life M of an atom in the substance is For the radioactive carbon
> The figure shows the sun located at the origin and the earth at the point (1, 0). (The unit here is the distance between the centers of the earth and the sun, called an astronomical unit: 1 AU ≈ 1.46 108 km.) There are five locations L1
> The average speed of molecules in an ideal gas is Where M is the molecular weight of the gas, R is the gas constant, T is the gas temperature, and is the molecular speed. Show that 3/2 "p'e-Mu/(2RT) du 4 M 2RT 8RT V TM
> Evaluate the definite integral. Fe4e dx/x√ ln x,
> Use Part 1 of the Fundamental Theorem of Calculus to find the derivative of the function. y = sin't dt - :
> A tank full of water has the shape of a paraboloid of revolution as shown in the figure; that is, its shape is obtained by rotating a parabola about a vertical axis. (a). If its height is 4 ft and the radius at the top is 4 ft, find the work required t
> If f∞-∞ f (x) dx is convergent and a and b are real numbers, show that L f(x) dx + [* f(x) dx = [°_f(x) dx + [° f(x) dx
> (a). Show that f∞-∞ x dx is divergent. (b). Show that This shows that we can’t define lim ,xd x dx = 0 Lf(x) dx = lim , f(x) dx
> Evaluate the definite integral. fa0 x√a2 – x2, dx
> A hole of radius is bored through the middle of a cylinder of radius R > r at right angles to the axis of the cylinder. Set up, but do not evaluate, an integral for the volume cut out.
> Calculate the moments Mx and My and the center of mass of a lamina with the given density and shape. : p= 2 quarter-circle
> Calculate the moments Mx and My and the center of mass of a lamina with the given density and shape. р3 10 y4 (4, 3)
> Sketch the region bounded by the curves, and visually estimate the location of the centroid. Then find the exact coordinates of the centroid. у%3 1/х, у— 0, х—D 1, х%3D2
> Sketch the region bounded by the curves, and visually estimate the location of the centroid. Then find the exact coordinates of the centroid. y = e*, y= 0, x= 0, x= 1
> Sketch the region bounded by the curves, and visually estimate the location of the centroid. Then find the exact coordinates of the centroid. Зх + 2у 3 6, у%3D0, х%3D0
> Use Part 1 of the Fundamental Theorem of Calculus to find the derivative of the function. tan x y = Jo
> Sketch the region bounded by the curves, and visually estimate the location of the centroid. Then find the exact coordinates of the centroid. y = 4 - x², y = 0
> The masses mi are located at the points Pi. Find the moments Mx and My and the center of mass of the system. %3D %3D P.(1, - 2), Р:(3, 4), Р:(-3, —7), Р.(6, — 1)
> The masses mi are located at the points Pi. Find the moments Mx and My and the center of mass of the system. т — 6, т — 5, ms — 10; P.(1,5), Р.(3, —2), Р.{(-2, —1)
> Point-masses mi are located on the x-axis as shown. Find the moment M of the system about the origin and the center of mass x. m,= 25 ++ -2 m3= 10 ++++ 7 m2 = 20 ++ 3
> A vertical, irregularly shaped plate is submerged in water. The table shows measurements of its width, taken at the indicated depths. Use Simpson’s Rule to estimate the force of the water against the plate. Depth (m) 2.0 2.5 3.0 3.
> A vertical dam has a semicircular gate as shown in the figure. Find the hydrostatic force against the gate. 2 m water level 12 m 4 m
> Find the volume of the described solid S. The base of S is an elliptical region with boundary curve 9x2 + 4y2 = 36. Cross-sections perpendicular to the -axis are isosceles right triangles with hypotenuse in the base.
> A large tank is designed with ends in the shape of the region between the curves y = 1/2x2 and y = 12, measured in feet. Find the hydrostatic force on one end of the tank if it is filled to a depth of 8 ft with gasoline. (Assume the gasoline’s density is
> A trough is filled with a liquid of density 840 kg/m3. The ends of the trough are equilateral triangles with sides 8 m long and vertex at the bottom. Find the hydrostatic force on one end of the trough.
> Sketch the region in the xy-plane defined by the inequalities x – 2y2 > 0, 1 – x – |y| > 0 and find its area.
> Use Part 1 of the Fundamental Theorem of Calculus to find the derivative of the function. h(x) = " VT+ r³ dr
> Use cylindrical shells to find the volume of the solid. The solid torus of Exercise 45 in Section 6.2. Exercise 45 in Section 6.2: (a). Set up an integral for the volume of a solid torus (the donut-shaped solid shown in the figure) with radii r and R.
> Find the volume of the described solid S. A pyramid with height and rectangular base with dimensions a and 2b.
> Let T be the triangular region with vertices (0, 0), (1, 0), (1, 2), and let V be the volume of the solid generated when T is rotated about the line x = a, where a > 1. Express a in terms of V.
> A vertical plate is submerged (or partially submerged) in water and has the indicated shape. Explain how to approximate the hydrostatic force against one side of the plate by a Riemann sum. Then express the force as an integral and evaluate it. -2 m
> A vertical plate is submerged (or partially submerged) in water and has the indicated shape. Explain how to approximate the hydrostatic force against one side of the plate by a Riemann sum. Then express the force as an integral and evaluate it. 5 m
> Show that the total length of the ellipse x = a sin θ, y = b cos θ, a > b > 0, is Where e is the eccentricity of the ellipse (e = c/a, where c = √a2 - b2). (T/2 L = 4a Jo VT - e² sin²0 do
> A tank is 8 m long, 4 m wide, 2 m high, and contains kerosene with density 820 kg/m3 to a depth of 1.5 m. Find (a) the hydrostatic pressure on the bottom of the tank, (b) the hydrostatic force on the bottom, and (c) the hydrostatic force on one end of th
> An aquarium 5 ft long, 2 ft wide, and 3 ft deep is full of water. Find (a) the hydrostatic pressure on the bottom of the aquarium, (b) the hydrostatic force on the bottom, and (c) the hydrostatic force on one end of the aquarium.
