# AP Test Review Help - Area of Region

• Apr 3rd 2011, 12:08 PM
eHovick
AP Test Review Help - Area of Region
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.
• Apr 3rd 2011, 12:46 PM
skeeter
Quote:

Originally Posted by eHovick
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.

make a sketch of $\displaystyle y = 2\cos{x}$.

this should get you started ...

area, $\displaystyle \displaystyle R = \int_0^{\frac{\pi}{2}} 2\cos{x} \, dx$
• Apr 3rd 2011, 01:29 PM
NOX Andrew
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 $\displaystyle y = 2\cos x$, such as one created by WolframAlpha. The other endpoint is the first intersection of the graph of $\displaystyle y = 2\cos x$ 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 $\displaystyle x = \dfrac{\pi}{2}$.

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.