Let R be the region in the first quadrant enclosed by the graph of y=2cosx, the x-axis, and the y-axis.
A) Find the area of the region R.
B) If the line x=a divides the region R into two regions of equal area, find a.
C) Find the volume of the solid obtained by revolving region R about the x-axis.
D) If R is the base of a solid whose cross sections perpendicular to the x-axis are semi-circles, find the volume of the solid.
I've been having trouble with problems like these... I can never figure out how to find the area and volume of these figures with so many lines.
First, identify the endpoints of the region R. One of the endpoints is implied by the definition of R: "Let R be the region in the first quadrant enclosed by... the y-axis." The y-axis is the same as the line x = 0, so one of the endpoints of the region R is the line x = 0.
The other endpoint can be found by examining a graph of , such as one created by WolframAlpha. The other endpoint is the first intersection of the graph of and the x-axis (y = 0) in the first quadrant ("Let R be the region in the first quadrant..."). According to the graph, the other endpoint is the line .
Now that you have a better understanding of the region R, you are better prepared to solve the problems.
By the way, I have taken both the AP Calculus AB and the AP Calculus BC examinations and I have helped AP Calculus AB students for 2 years, so I have a modest knowledge of the exam. I would focus on parts (a), (c), and (d) if I were you. Questions similar to those parts, especially parts (a) and (c), are common. However, I have never seen part (b) on the exam, so I wouldn't worry too much about it.