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(e) The region (see Fig. 33-17) inside the circle x 2 + y2 = r 2 , with 0 x a < r; about the y-axis. (This gives the volume cut from a sphere of radius r by a pipe of radius a whose axis is a diameter of the sphere.) (f) The region (see Fig. 33-18) inside the circle x 2 + y2 = r 2 , with x 0 and y 0, and above the line y = a, where 0 a < r; about the y-axis. (This gives the volume of a polar cap of a sphere.) (g) The region bounded by y = 1 + x 2 and y = 5; about the x-axis. (h) The region (see Fig. 33-19) inside the circle x 2 + (y b)2 = a2 , with 0 < a < b, about the x-axis. [Hint: When you
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obtain an integral of the form
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gives the volume of a doughnut-shaped solid. (i) The region bounded by x 2 = 4y and y = x/2; about the y-axis. (j) The region bounded by y = 4/x and y = (x 3)2 ; about the x-axis. (Notice that the curves intersect when x = 1 and x = 4. What is special about the intersection at x = 1 ) (k) The region of part (j); about the y-axis. (l) The region bounded by xy = 1, x = 3, y = 0; about the x-axis. (m) The region of part (l); about the y-axis.
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a2 x 2 dx notice that this is the area of a semicircle of radius a.] This problem
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Fig. 33-17
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Fig. 33-18
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Fig. 33-19
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33.9 Use the cross-section formula to nd the volume of the following solids. (a) The solid has a base which is a circle of radius r. Each cross section perpendicular to a xed diameter of the circle is an isosceles triangle with altitude equal to one-half of its base. (b) The solid is a wedge, cut from a perfectly round tree of radius r by two planes, one perpendicular to the axis of the tree and the other intersecting the rst plane at an angle of 30 along a diameter (see Fig. 33-20). (c) A square pyramid with a height of h units and a base of side r units. [Hint: Locate the x-axis as in Fig. 33-21. By similar right triangles, h x d = e h and which determines A(x).] (d) The tetrahedron (see Fig. 33-22) formed by three mutually perpendicular faces and three mutually perpendicular edges of lengths a, b, c. [Hint: Another pyramid; proceed as in part (c).] A(x) = r2 d 2 e
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Fig. 33-20
Fig. 33-21
Fig. 33-22
33.10 (a) Let R be the region between x = 0 and x = 1 and bounded by the curves y = x 3 and y = 2x. Find the volume of the solid obtained by revolving R about the y-axis. (b) Let R be the region between the curves y = 2x x 2 and y = 1 x. Find 2 the volume of the solid obtained by revolving R about the y-axis. 33.11 Let R be the region in the rst quadrant bounded by y = x 3 + x, x = 2, and the x-axis. (a) Find the volume of the solid obtained by revolving R about the line y = 3. (b) Find the volume of the solid obtained by revolving R about the line x = 1. 33.12 Let R be the region in the rst quadrant bounded by x = 4 y2 and y2 = 4 2x. (a) Sketch R. (b) Find the volume of the solid obtained by revolving R about: (i) the x-axis; (ii) the y-axis. 33.13 Let R be the region in the second quadrant bounded by y = 2x 2 , y = x 2 + x + 2, and the y-axis. (a) Sketch R (b) Find the volume of the solid obtained by revolving R about the y-axis. 33.14 Let R be the region in the second quadrant bounded by y = 1 + x 2 and y = 10. (a) Sketch R. Then nd the volume of the solid obtained by revolving R about: (b) the x-axis; (c) the y-axis; (d) the line y = 1; (e) the line x = 1.
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