MTH 252: Spring 2017 Week 4 Worksheet Do all the problems without using a calculator. You must show all the steps. 1. a. Find the points of intersection for two curves y = −x and y = 2−x 2 . Between the points of intersection, which one of the two curves is above the other? b. Find the points of intersection for two curves y = 4 1 + x 2 and y = 1. Between the points of intersection, which one of the two curves is above the other? c. Find the points of intersection for two curves y = √ x + 1 and y = 1 x + 1 . For x to the right of the intersection point, which of the two curves is above the other? 1 2. Let f(x) = √ x and g(x) = x 4 . We have that f(x) ≥ g(x) over [0, 1]. a. Compute the area below the graph of f(x) over [0, 1]. b. Compute the area below the graph of g(x) over [0, 1]. c. Using the answers from a and b, compute the area of the region between f(x) and g(x) (the part below f(x) and above g(x)) over [0, 1]. d. In general, if f(x) ≥ g(x) over [a, b], express the area between the two curves over [a, b] as a single definite integral. 2 3. Suppose that we have two concentric circles where the radius of the larger circle is given by f(x) and the radius of the smaller by g(x). Compute the area of the region between the two circles. (The shaded region in the figure below.) 4. Suppose that we have a hollow cylinder without the top or bottom whose height is given by the function f(r) where r is the radius. If we cut open the cylinder and flatten out as a rectangle as shown in the figure below, what is the area of the rectangle? 3

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