Math 1A: Discussion ExercisesGSI: Theo Johnson-Freydhttp://math.berkeley.edu/∼theojf/09Spring1A/Find two or three classmates and a few feet of chalkboard. As a group, try your hand at thefollowing exercises. Be sure to discuss how to solve the exercises — how you get the solution ismuch more important than whether you get the solution. If as a group you agree that you allunderstand a certain type of exercise, move on to later problems. You are not expected to solve allthe exercises: in particular, the last few exercises may be very hard.Many of the exercises are from Single Variable Calculus: Early Transcendentals for UC Berke-ley by James Stewart; these are marked with an §. Others are my own, or are independently marked.Volumes1. § Sketch the following curves and the region enclosed. Find the volume of the solid formedby rotating the region around the given line.(a) y =14x2, y = 5 − x2;about the x-axis.(b) x = y2, x = 2y;about the y-axis.(c) y = x, y =√x;about y = 1.(d) y = 1 + sec x, y = 3;about y = 1.(e) x = y2, x = 1;about x = 1.(f) y = x2, x = y2;about x = −1.2. Prove the following theorem of Archimedes:A cylinder with the same height and diameter has the same total volume as the total volume ofthe sphere with the same diameter together with the cone with the same height and diameter.3. § Find the volume of the right circular cone with height h and base r.4. § Find the volume of the pyramid with height h and rectangular base with dimensions b ×2b.5. § Find the volume of the tetrahedron with three mutually perpendicular faces and threemutually perpendicular edges with lengths 3 cm, 4 cm, and 5 cm.6. Use calculus to prove that V =13Bh, where V is the volume of a “pyradmid” whose base issome arbitrary shape of area B, whose height is h, and all of whose cross sections are similarand shrink linearly from the base to the vertex.7. § Find the volume of the solid doughnut. This is the volume of revolution formed by rotatinga circle of radius r centered at (R, 0) around the y-axis, where R > r > 0.8. § Find the volume of the solid whose base is the region enclosed by the parabola y = 1 − x2and the x-axis, and whose cross-sections perpendicular to the y-axis are squares.9. (a) § Two cylinders with the same radius intersect at right angles. Find the volume of theintersection.(b) Three cylinders with the same radius intersect at right angles. Find the volume of theintersection.10. § A bead is made by boring a cylindrical whole of radius r through a solid sphere of radiusR (where 0 < r < R). Find the volume of material of the bead.11. § Two solids of revolution are formed by rotating the same region around different lines:namely, a region above the x-axis is rotated around the x-axis and around y = −k for someconstant k > 0. Prove that the difference between the volumes of the two solids depends onlyon k and on the area of the region, and find a formula for this dependence.12. § A sphere of radius 1 intersects a smaller sphere of radius r < 1 in such a way that theirintersection is a circle of radius r. In other words, they intersect in a great circle of the smallsphere. Find r so that the volume inside the small sphere and outside the large sphere is aslarge as
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