Evaluate the definite integral. f1/21/6 csc πt cot πt dt
> Suppose that for the tank in Exercise 19 the pump breaks down after 4.7 × 105 J of work has been done. What is the depth of the water remaining in the tank? Exercise 19: A tank is full of water. Find the work required to pump the water out
> Evaluate the integral. f ln 3√x dx
> Evaluate the integral. f x2 cos mx dx
> Find a positive continuous function f such that the area under the graph of f from 0 to t is A (t) = t3 for all t > 0.
> Evaluate the integral. f x2 sin πx dx
> Evaluate the integral. f t sin 2t dt
> Evaluate the integral. f rer/2 dr
> Evaluate the integral. f xe-x dx
> (a). Prove the reduction formula f cosnx dx = 1/n cosn-1 x sin x + n – 1/n f cosn-2x dx (b). Use part (a) to evaluate f cos2 x dx. (c). Use parts (a) and (b) to evaluate f cos4x dx.
> (a). Use the reduction formula in Example 6 to show that f sin2x x dx = x/2 – sin2x/4 + C. (b). Use part (a) and the reduction formula to evaluate f sin4x dx.
> Evaluate the indefinite integral. Illustrate, and check that your answer is reasonable, by graphing both the function and its antiderivative (take C = 0). (x? sin 2x dx
> (a). Show that the volume of a segment of height of a sphere of radius r is (See the figure.) (b). Show that if a sphere of radius 1 is sliced by a plane at a distance from the center in such a way that the volume of one segment is twice the volume of
> Evaluate the indefinite integral. Illustrate, and check that your answer is reasonable, by graphing both the function and its antiderivative (take C = 0). VI +x² dx
> Evaluate the indefinite integral. Illustrate, and check that your answer is reasonable, by graphing both the function and its antiderivative (take C = 0). x3/2 In x dx
> Evaluate the indefinite integral. Illustrate, and check that your answer is reasonable, by graphing both the function and its antiderivative (take C = 0).
> (a). Find the number such that the line x = a bisects the area under the curve y = 1/x2, 1 < x < 4. (b). Find the number such that the line y = b bisects the area in part (a).
> First make a substitution and then use integration by parts to evaluate the integral.
> Evaluate the integral. f x cox x dx
> First make a substitution and then use integration by parts to evaluate the integral.
> First make a substitution and then use integration by parts to evaluate the integral.
> First make a substitution and then use integration by parts to evaluate the integral.
> First make a substitution and then use integration by parts to evaluate the integral.
> (a). Graph several members of the family of functions f (x) = (2cx – x2)/c3 for c > 0 and look at the regions enclosed by these curves and the -axis. Make a conjecture about how the areas of these regions are related. (b). Prove your conjecture in part (
> Evaluate the integral. ft0 ex sin (t – s) ds
> Evaluate the integral. f21 (ln x)2 dx
> Evaluate the integral. f10 r3/√4 + r2, dr
> Evaluate the integral. f1/20 cos-1 x dx
> (a). If g (x)= (sin2x)/x2, use your calculator or computer to make a table of approximate values of ft1g (x) dx for t = 2, 5, 10, 100, 1000, and 10,000. Does it appear that f∞1g (x) dx is convergent? (b). Use the Comparison Theorem with f (x) = 1/x2 to s
> Evaluate the integral. F√31 arctan (1/x) dx
> Evaluate the integral using integration by parts with the indicated choices of u and dv.
> Evaluate the integral. f10 y/e2y dy
> Evaluate the integral. f94 ln y/√y dy
> Evaluate the integral. f21 ln x/x2 dx
> Three mathematics students have ordered a 14-inch pizza. Instead of slicing it in the traditional way, they decide to slice it by parallel cuts, as shown in the figure. Being mathematics majors, they are able to determine where to slice so that each gets
> Evaluate the integral. f10 (x2 + 1) e-x dx
> Evaluate the integral. fπ0 t sin 3t dt
> Evaluate the integral. f e-θ cos 2θ dθ
> Evaluate the integral. f e2θ sin 3θ dθ
> Evaluate the integral. f sin-1x dx
> Find the area of the region bounded by the parabola y = x2, the tangent line to this parabola at (1, 1) and the x-axis.
> Evaluate the integral. f p5 ln p dp
> Evaluate the integral using integration by parts with the indicated choices of u and dv.
> Evaluate the indefinite integral. f (3x – 2)20 dx
> Evaluate the indefinite integral. f x2 (x3 + 5)9 dx
> The Fresnel function S (x) = fx0sin (1/2Ï€t2) dt was introduced in Section 5.4. Fresnel also used the function in his theory of the diffraction of light waves. (a). On what intervals C is increasing? (b). On what intervals is C concave upward
> Evaluate the indefinite integral. f x sin (x2) dx
> Evaluate the integral by making the given substitution. f sec2(1/x)/x2 dx, u = 1/x
> Evaluate the integral by making the given substitution. f cos3θ sin θ d θ, u = cos θ
> Evaluate the definite integral. F41 e√x/√x dx
> Evaluate the definite integral. f10 x2 (1 + 2x3)5 dx
> A swimming pool is 20 ft wide and 40 ft long and its bottom is an inclined plane, the shallow end having a depth of 3 ft and the deep end, 9 ft. If the pool is full of water, estimate the hydrostatic force on (a) the shallow end, (b) the deep end, (c) on
> Evaluate the definite integral. f√π0 x cos (x2) dx
> Evaluate the definite integral. f10 3√1 + 7x, dx
> Evaluate the definite integral. f10 (3t – 1) dt
> If a projectile is fired with an initial velocity v at an angle of inclination θ from the horizontal, then its trajectory, neglecting air resistance, is the parabola (a). Suppose the projectile is fired from the base of a plane that is inc
> Use the graphs of f, f', f" to estimate the -coordinates of the maximum and minimum points and inflection points of f.
> Evaluate the definite integral. f10 cos (πt/2) dt
> Verify by differentiation that the formula is correct. f x cos x dx = x sin x + cos x + C
> Evaluate the integral by making the given substitution. f dt/ (1 – 6t)4, u = 1 – 6t
> Verify by differentiation that the formula is correct. f cos3 x dx = sin x - 1/3 sin3 x + C
> Evaluate the indefinite integral. Illustrate and check that your answer is reasonable by graphing both the function and its antiderivative (take C = 0).
> Evaluate the indefinite integral. Illustrate and check that your answer is reasonable by graphing both the function and its antiderivative (take C = 0). //
> Evaluate the indefinite integral. f x/1 + x4 dx
> Suppose you make napkin rings by drilling holes with different diameters through two wooden balls (which also have different diameters). You discover that both napkin rings have the same height h, as shown in the figure. (a). Guess which ring has more
> Evaluate the indefinite integral. f 1 + x/1 + x2 dx
> Evaluate the indefinite integral. f sin x/1 + cos2x dx
> A rectangular beam will be cut from a cylindrical log of radius 10 inches. (a). Show that the beam of maximal cross-sectional area is a square. (b). Four rectangular planks will be cut from the four sections of the log that remain after cutting the squ
> Evaluate the indefinite integral. f sin2x/1 + cos2x dx
> Evaluate the indefinite integral. f ex/ex + 1, dx
> Evaluate the indefinite integral. f x(2x + 5)8 dx
> Evaluate the indefinite integral. f x2 √2 + x dx
> Evaluate the integral by making the given substitution. f x2 √x3 + 1 dx, u = x3 + 1
> Evaluate the indefinite integral. f sec3x tan x dx
> Evaluate the indefinite integral. f dt/cos2t√1 + tan t
> Evaluate the indefinite integral. f dx/√1 – x2, sin-1x
> Find the area bounded by the loop of the curve with parametric equations x = t2, y – t3 – 3t.
> Evaluate the indefinite integral. f cos (π/x)/x2 dx
> Investigate the family of curves given by f (x) = x4 + x3 + cx2 In particular you should determine the transitional value of c at which the number of critical numbers changes and the transitional value at which the number of inflection points changes. Il
> Evaluate the indefinite integral. f √cot x, csc2x dx
> Evaluate the indefinite integral. f sin (ln x)/x dx
> Evaluate the indefinite integral. f (x2 + 1) (x3 + 3x)4 dx
> Evaluate the indefinite integral. f tan-1x/1 + x2 dx
> Evaluate the indefinite integral. f cos x/sin2x, dx
> Evaluate the indefinite integral. f sec 2θ tan 2θ dθ
> Evaluate the integral by making the given substitution. f x3 (2 + x4)5 dx, u = 2 + x
> Evaluate the indefinite integral. f ex √1 + ex dx
> Evaluate the indefinite integral. f z2/z3 + 1 dz
> 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.
> (a). If f (x) = 0.1 ex + sin x, -4 < x < 4 use a graph of f to sketch a rough graph of the antiderivative F of f that satisfies F (0) = 0. (b). Find an expression for F (x). (c). Graph F using the expression in part (b). Compare with your sketch in part
> Evaluate the indefinite integral. f a + bx2 / √3ax + bx3 dx
> Evaluate the indefinite integral. f sin √x/ √x dx
> Evaluate the indefinite integral. f dx/ 5 – 3x
> Evaluate the indefinite integral. f x/ (x2 + 1)2 dx
> Evaluate the indefinite integral. f (ln x)2/x dx
> Evaluate the indefinite integral. f ex cos (ex) dx
> Evaluate the indefinite integral. f sin πt dt
> Evaluate the indefinite integral. f (3t + 2)2.4 dt
> Evaluate the integral by making the given substitution. f e-x dx, u = -x
> Use Part 1 of the Fundamental Theorem of Calculus to find the derivative of the function.
> (a). How would you evaluate fx5e-2x dx by hand? (Don’t actually carry out the integration.) (b). How would you evaluate fx5e-2x dx using tables? (Don’t actually do it.) (c). Use a CAS to evaluate fx5e-2x dx. (d). Graph the integrand and the indefinite in
