A region R is defined as the area in the first quadrant bounded by the curves y = x² and y = x³         (a)    What is the area of R?
        (b)    What volume is generated when R is rotated about the line y = 2?
        (c)    If a solid has R as its base and semicircular cross-sections perpendicular to the x-axis, find its volume.

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(a)    The region R is pictured below. The blue graph is the square and the red is the cubic. Thet intersect at x = 1, so the area (top minus bottom when everything is in terms of x) will be found with: (b)    Because the rotational volume will not be completely solid (there's a hole), you have to use the Washer Method. The outer radius is R(x) = x²(2) = x² + 2. The inner radius is r(x) = x³ + 2. (c)    The expression x² x³ represents the general diameter of a cross-section. Remember, to find the volume indicated, you have to integrate the area of one cross section. Since the cross-sections are semicircles, you integrate A = (r²)/2. The radius will be (x² x³)/2:

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AP Calculus BC: Integration By Partial Fractions Review

Integrate: 

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This problem is a good candidate for integration by parts since the denominator is factorable--that is your cue for this technique.