Problem+of+the+Week


 * Week 1

A graphing calculator is required for some problems or parts of problems. **

1. Let the shaded region be bounded by the following functions: f(x)=-x+1 ; g(x)=e x ; h(x)=-x+4 ; j(x)=ln(x) …………………………………………………

(a) Find the area of the shaded region.

(b) Write, but do not evaluate, an integral expression that can be used to find the volume of the solid generated when the shaded region in part (a) is revolved about the line y=3.

(c) Find the equation of the line perpendicular to the line y=x that divides the shaded region into two equal parts.

** Week 2

A graphing calculator is NOT allowed in this section! ** 1. Let the shaded regions be bounded by the function y=x (1/2) and the lines y=0, y=1, y=2, and y=3 on the interval from x=0 to x=9 as shown in the figure below.



(a) Find the total area of the shaded regions on the interval from x=0 to x=9.

(b) Find the volume of the solid generated when the shaded region on the interval x=4 to x=9 is revolved about the x axis.

(c) Find the volume of the solid generated when the shaded region on the interval x=4 to x=9 is revolved about the y axis.

** Week 3

A graphing calculator is required for some problems or parts of problems. ** 1. Let ** R ** be the shaded region bounded by the functions g(x) and f(x). Let ** S ** be the unshaded region bounded by the functions g(x), f(x) , and h(x) as shown in the figure below. Note: f(x) = x (1/2) ; g(x) = x 2 ; h(x) = green line that intersects f(x) and g(x) as shown.

(a) Find the area of the region R in the figure above.

(b) What is the equation of the line, h(x), perpendicular to y = x such that the shaded region R is equal to the unshaded region S?

(c) Write, but do not evaluate, an integral expression that can be used to find the volume of the solid generated when the unshaded region S is revolved about the x axis.

** Week 4

A graphing calculator is required for some problems or parts of problems. ** 1. Let ** R ** be the shaded region bounded by the function f(x), the x-axis, and the circle x 2 + y 2 = 2 as shown in the figure below.



(a) Find the area of the shaded region R in the figure above.

(b) What is the equation of the line, h(x), perpendicular to y = x that divides the shaded region R into two equal parts.

(c) Write, but do not evaluate, an integral expression that can be used to find the volume of the solid generated when the shaded region R is revolved about the x axis.

** WEEK 5 **


 * A graphing calculator is NOT allowed in this section! **

Given: a circle centered at the origin. **
 * 1. ****
 * Given: the shaded region T is equal to the shaded region S. **
 * Given: h(x) = -x + c, **



Find the value of C in terms of r, the radius of the given circle. **
 * a) ****

generated by revolving the shaded region T about the X axis. ** (c-e) Given: a circle with a radius of 2. **
 * b) **
 * Find the volume (in terms of r, the radius of the given circle)

Find the equation of the line h(x). **
 * c)

Find the volume generated by revolving the region T about the X axis. **
 * d)

Write, but do not evaluate the integral expression that represents the volume generated by revolving the region S about the X axis. **
 * e)