Questions from General Calculus

Q: Let R be the region that lies between the curves y =

Let R be the region that lies between the curves y = xm and y = xn, 0 < x < 1, where m and n are integers with 0 < n < m. (a). Sketch the region R. (b). Find the coordinates of the centroid of R. (c)....

Q: Shown is the graph of traffic on an Internet service provider’s T1

Shown is the graph of traffic on an Internet service provider’s T1 data line from midnight to 8:00 AM. D is the data throughput, measured in megabits per second. Use Simpsonâ&#...

Q: (a). Use the table of integrals to evaluate F (

(a). Use the table of integrals to evaluate F (x) = f f (x) dx, where f (x) = 1/x√1 – x2 What is the domain of f and F? (b). Use a CAS to evaluate F (x). What is the domain of the function F that the...

Q: A uniform disk with radius 1 m is to be cut by

A uniform disk with radius 1 m is to be cut by a line so that the center of mass of the smaller piece lies halfway along a radius. How close to the center of the disk should the cut be made? (Express...

Q: The figure shows a pendulum with length L that makes a maximum

The figure shows a pendulum with length L that makes a maximum angle θ0 with the vertical. Using Newton’s Second Law, it can be shown that the period T (the time for one...

Q: The intensity of light with wavelength traveling through a diffraction grating with

The intensity of light with wavelength traveling through a diffraction grating with N slits at an angle θ is given by I (θ) = N2 sin2k/k2, where k = (πNd sin θ)/λ and d is the distance between adjacen...

Q: Sketch the graph of a continuous function on [0, 2

Sketch the graph of a continuous function on [0, 2] for which the right endpoint approximation with n = 2 is more accurate than Simpson’s Rule.

Q: Sketch the graph of a continuous function on [0, 2

Sketch the graph of a continuous function on [0, 2] for which the Trapezoidal Rule with n = 2 is more accurate than the Midpoint Rule.

Q: Use the Trapezoidal Rule with n = 10 to approximate f200 cos

Use the Trapezoidal Rule with n = 10 to approximate f200 cos (πx) dx. Compare your result to the actual value. Can you explain the discrepancy?

Q: Use integration by parts to prove the reduction formula. f

Use integration by parts to prove the reduction formula. f (ln x) n dx = x (ln x) n – n f (ln x) n-1 dx