Math 44 Activity 9 (Due by end of class August 6). Find the value of c, c, that minimizes the volume of the solid generated by revolving the region between the graphs of y 4 and y c from to about the line y c.. The curve defined by y 5 of the solid obtained by revolving the region inside the loop around: a) the -ais..} {Hint: 4 V c c d includes the loop shown in the graph. Find the volume b) the y-ais. c) the line 5.
3. A rectangle R of length 4 and width 3 with one of its diagonals along the -ais is revolved about the -ais. Find the volume of the resulting solid. 3 4 h 5 h 9 h 5 6 4. The region between the graphs of the functions y k, k and y sin from to is to be revolved about the line y k. y sin y k a) Determine the eact value of k so that the volume of the resulting solid is minimum. V k k d.} {Hint: sin b) Determine the eact value of k so that the volume of the resulting solid is maimum.
5. A gasoline tank is an ellipsoid of revolution formed by revolving the region bounded by the graph of y about the y-ais. Find the depth of 6 9 the gasoline in the tank when it is filled to ¼ its capacity. depth 6. Suppose that f is a continuous function with the property that, for every a >, the volume swept out by revolving the region enclosed by the -ais and the graph of f from to 3 a is a f.. Find all possible functions, 7. Find the number c so that the areas of the shaded regions are equal. {Hint:If the areas are equal, d 3 8 7 c d and 8d 7d c.} 3 then y 8 7 3 y c d
sin ;. f is continuous everywhere and has the property that f sin. ; Find the volume of the solid generated by revolving the region below the graph of f from to about the y-ais. 8. Let f 9. A wedge is cut from a right circular cylinder of radius r. The upper surface of the wedge is in a plane through a diameter of the circular base and makes an angle with the base. Find the volume of the wedge. y r
. For c, the graphs of y c and y c bound a plane region. Revolve this region around the horizontal line y to form a solid. For what value of c is the volume of this c solid maimal? Minimal? c c c c. Evaluate sin d by interpreting it as an area and integrating with respect to y instead of. {Hint: sin d sin y dy.}
. a) Find the area of the region between the graphs of y and the circle y k 5, by first finding the value of k that makes the circle tangent to the graph of y. {Hint: This can be done without calculus.} b) Find the volume when this region is revolved about the y-ais. c) Find the volume when this region is revolved about the -ais. 3. a) Find the area of the region inside the circle y and above the line y. b) Find the volume when this region is revolved about the y-ais. c) Find the volume when this region is revolved about the -ais.
4. Consider the ellipse 4y 4. Let R be the region in the first quadrant bounded by the ellipse, the y-ais and the line y 3. Let R be the region in the first quadrant bounded by the ellipse, the -ais and the line y m. What does m have to be for R and R to have the same area? R R {Hint: 37 4m 4 d and area of m area of R 3 m R y y d.} 5. Let R be the region bounded by the parabola y and the -ais. Find the equation of the line y m ; m that divides R into two regions of equal areas. m m d d.} {Hint:
6. A torus is formed by revolving the region bounded by the circle y-ais. Calculate its volume. y about the {Hint: y y dy or d.} 3
7. Water partially fills a cylindrical bucket of radius a feet and height L feet. When the bucket is rotated about its central ais with angular speed of radians/sec, the surface of the water assumes a parabolic shape whose profile is given by y H ; a a, where g g 9.8 m. sec a) For a fied volume of water, V cubic feet, how does H depend on? L a H WATER a a { V H d, so solve for H in terms of.} g b) At what value of does the water surface touch the bottom of the bucket? {For what value of, does H equal zero?} c) At what value of does water spill over the top of the bucket? {Water spills over the top of the bucket, when the height at the edge eceeds L. The height at the edge is given by H a, and you already have a formula for H.} g
8. Consider the region in the first quadrant bounded by y, y,, and b b. a) If V is the volume generated by revolving the region about the -ais, then find a formula for V. b) If V y is the volume generated by revolving the region about the y-ais, then find a formula for V y. c) Is there a value of b so that V Vy? If so, what is it?
9. a) Find the area of the bounded region in the first quadrant between the curves y. y y and y b) Find the volume when this region is revolved about the -ais. c) Find the volume when this region is revolved about the y-ais.. a) Find the area of the bounded region R in the first quadrant, indicated by the graph, bounded by the curves y, y 4, and y 4 6. R b) Find the volume when this region is revolved about the -ais. c) Find the volume when this region is revolved about the y-ais.
. a) Set-up an integral(s) to find the area of the bounded region R in the first quadrant bounded by the curves y and sin y. d R c R a b b) Set-up an integral(s) to find the volume when this region is revolved about the -ais. c) Set-up an integral(s) to find the volume when this region is revolved about the y-ais.
. A cup is formed by revolving the curve y 3 ; about the y-ais. The cup is filled to the brim with water. If you plan to drink eactly half of the water, at what height should the water be when you stop drinking? h 8 {Hint: 3 3 y dy y dy.} 3. Use elementary geometry and symmetry to find the area of the region between the graphs 3 of f and g.
4. Consider a curve y f on,, and f point, f with that passes through the origin, with f increasing, differentiable area A above the curve. Find all the functions f that satisfy these conditions. Start by showing that f A f t dt. Or if you prefer,, so that when horizontal and vertical lines are each drawn from the to the coordinate aes, the area A under the curve is twice the A A f f t dt. f t dt and A A, f