MATH 101 ASSIGNMENT 3 1. In written assignments you are asked to solve difficult problems as a team. Understanding how to approach or begin a problem is often the most challenging aspect of problem solving. Discuss the following prompts together and write a one or two paragraph response addressing all three. (a) What strategies does your team use when you are stuck and do not know how to approach or begin to solve a problem? (b) Which strategies have you found to be the most effective? (c) Is the way you approach difficult problems in math similar to how you approach difficult problems in other subject areas? Explain why or why not. 2. In this question, we'll approximate the volume of a thin cylinder. The approximation may seem odd at first, but it will be useful later on. (a) Let x be a positive constant. Evaluate lim Ax->0 2TxAx T(x+Ax)2 - Tx2 . (b) Let R be a washer-shaped region with inner radius x and outer radius x + A. Give the area of R. Ax x Figure 1: region R (c) Let Ar be a positive number, close to 0. Give a geometric argument for why the area of R might be close to 2TxAx. (Hint: imagine "unrolling" the circle.) (d) Use part (a) to further justify why the area of R might be close to 2xAx when Ax is a positive number that is close to 0. (e) Imagine a thin cylinder whose base is R and height is y. Use the approximate area from (c) to give an approximate volume of the cylinder. T y Figure 2: thin cylinder
x +2 3. Let S be the solid formed by rotating the finite area bounded by x = 0, x = 3, y = 0, and y = x+1 around the y-axis. In this question, we'll find the volume of S using cylindrical shells1. y x +2 y = x + 1 x 3 Figure 3: the region rotated to form S (a) Sketch the portion of the solid resulting from rotating the finite area bounded by x = 1, x = 1.1, y = 0, and y = x + 2 x +1 around the y-axis. (b) Use your formula from 2(e) to approximate the volume of the solid in (a). (c) What is the approximate volume of a cylindrical shell with radius xo, width Axo, and height x0 + 2 x0 + 1 ? (Continue to use the formula from 2(e).) (d) We can imagine the solid S consisting of layers of thin cylinders, or cylindrical shells, similar to the shape you found in (a). y x + 2 x + 1 y = x 3 Figure 4: approximating S using cylinders We want to create a collection of these cylindrical shells that makes up all of S. What are the largest and smallest radii such shells could have? (e) "Adding up" the volumes of all the cylindrical shells will give us the volume of S.