Questions from General Calculus

Q: If f is continuous on R, prove that fbaf (x

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.

Q: If a and b are positive numbers, show that

If a and b are positive numbers, show that

Q: (a). If f is continuous, prove that fπ/

(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.

Q: A manufacturer of corrugated metal roofing wants to produce panels that are

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...

Q: Suppose h is a function such that h (1) =

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.

Q: The area labeled B is three times the area labeled A.

The area labeled B is three times the area labeled A. Express a in terms of b.

Q: Evaluate the limit by first recognizing the sum as a Riemann sum

Evaluate the limit by first recognizing the sum as a Riemann sum for a function defined on [0, 1].

Q: Evaluate the limit by first recognizing the sum as a Riemann sum

Evaluate the limit by first recognizing the sum as a Riemann sum for a function defined on [0, 1].

Q: Let S be the solid obtained by rotating the region shown in

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. D...

Q: Find the most general antiderivative of the function. (Check your

Find the most general antiderivative of the function. (Check your answer by differentiation.)