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Math Help - AP Test Review Help - Area of Region

  1. #1
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    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.
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  2. #2
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    Quote Originally Posted by eHovick View Post
    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 y = 2\cos{x}.

    this should get you started ...

    area, \displaystyle R = \int_0^{\frac{\pi}{2}} 2\cos{x} \, dx
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  3. #3
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    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 y = 2\cos x, such as one created by WolframAlpha. The other endpoint is the first intersection of the graph of 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 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.
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