Questions from General Calculus

Q: Evaluate the indefinite integral. Illustrate, and check that your answer

Evaluate the indefinite integral. Illustrate, and check that your answer is reasonable, by graphing both the function and its antiderivative (take C = 0).

Q: (a). Show that the volume of a segment of height

(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...

Q: Evaluate the indefinite integral. Illustrate, and check that your answer

Evaluate the indefinite integral. Illustrate, and check that your answer is reasonable, by graphing both the function and its antiderivative (take C = 0).

Q: (a). Use the reduction formula in Example 6 to show

(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.

Q: (a). Prove the reduction formula f cosnx dx = 1

(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.

Q: Evaluate the integral. f xe-x dx

Evaluate the integral. f xe-x dx

Q: Evaluate the integral. f rer/2 dr

Evaluate the integral. f rer/2 dr

Q: Evaluate the integral. f t sin 2t dt

Evaluate the integral. f t sin 2t dt

Q: Evaluate the integral. f x2 sin πx dx

Evaluate the integral. f x2 sin πx dx

Q: Find a positive continuous function f such that the area under the

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.