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Math 1320 - Lab 1 Name: For problems 1 through 3, draw a detailed picture. 1. Find the √ volume of the solid formed by rotating the finite region bounded by the graphs of y = 4 x, y = x4 around the x-axis. 2. Let D be the region enclosed by the graphs y = x2 and y = 8 − x2 . Find the volume of solid generated by rotating D about y = −1. 3. A machinist has a hemisphere with radius 1 cm made out of some alloy metal. Suppose they wish to have a volume of exactly 2 cm3 of this metal. How big of a hole would they have to bore through the center of the hemisphere to achieve this volume? Page 2 4. Find the volume of a pyramid of height h whose base is an equilateral triangle of length L by following the following individual steps/hints: a. We can compute the volume of this object just as we would any other object: Z h A(y) dy, 0 where A(y) is the cross sectional area in the following figure: Thus, we must determine A(y). To do so, first consider the cross section shown with width s. What is the area of this cross section? Hint: it is also an equilateral triangle. Page 3 b. Now that we have the area for a particular width s, relate s and y to L and h by noting that similar triangles occur. c. From this, obtain an explicit expression for A(y), involving only L, h, y but not s. d. Compute the volume of the shape. Page 4