find the volume of the solids revolving the triangle vertice

Solution

12. Suppose that instead of completing a pyramid, the builders at Gizeh had stopped at height 250 feet (with a square plateau top of side 375 feet). Compute the volume of this structure. Explain why the volume is greater than half the volume of the pyramid in exercise 11. 13. A church steeple is 30 feet tall with square cross sections. The square at the base has side 3 feet, the square at the top has side 6 inches, and the side varies linearly in between. Compute the volume. 14. A house attic has rectangular cross sections parallel to the ground and triangular cross sections perpendicular to the ground. The rectangle is 30 feet by 60 feet at the bottom of the attic and the triangles have base 30 feet and height 10 feet. Compute the volume of the attic. 15. A pottery jar has circular cross sections of radius inches for 0 x 2. Sketch a picture of the jar and compute its volume. 16. A pottery jar has circular cross sections of radius inches for 0 x 2. Sketch a picture of the jar and compute its volume. 17. Suppose an MRI scan indicates that cross-sectional areas of adjacent slices of a tumor are as given in the table. Use Simpson\'s rule to estimate the volume. x(cm) 0.0 0.1 0.2 0.3 0.4 0.5 A(x) (cm2) 0.0 0.1 0.2 0.4 0.6 0.4 x 0.6 0.7 0.8 0.9 1.0 A(x) (cm2) 0.3 0.2 0.2 0.1 0.0 18. Suppose an MRI scan indicates that cross-sectional areas of adjacent slices of a tumor are as given in the table. Use Simpson\'s rule to estimate the volume. x (cm) 0.0 0.2 0.4 0.6 0.8 1.0 1.2 A(x) (cm2) 0.0 0.2 0.3 0.2 0.4 0.2 0.0 19. Estimate the volume from the cross-sectional areas. x (ft) 0.0 0.5 1.0 1.5 2.0 A(x) (ft2) 1.0 1.2 1.4 1.3 1.2 20. Estimate the volume from the cross-sectional areas. x (m) 0.0 0.1 0.2 0.3 0.4 A(x) (m2) 2.0 1.8 1.7 1.6 1.8 x (m) 0.5 0.6 0.7 0.8 A(x) (m2) 2.0 2.1 2.2 2.4 In exercises 21-28, compute the volume of the solid formed by revolving the given region about the given line. 21. Region bounded by y = 2-x, y = 0 and x = 0 about (a) the xaxis; (b) y = 3 22. Region bounded by y = x2, y = 0 and x = 2 about (a) the xaxis; (b) y = 4 23. Region bounded by y = , y = 2 and x = 0 about (a) the yaxis; (b) x = 4 24. Region bounded by y = 2x, y = 2 and x = 0 about (a) the yaxis; (b) x = 1 25. Region bounded by y = ex, x = 0, x = 2 and y = 0 about (a) the yaxis; (b) y = -2. Estimate numerically. 26. Region bounded by y = cos x, x = 0 and y = 0 about (a) y = 1 ; (b) y = -1 27. Region bounded by y = x3, y = 0 and x = 1 about (a) the yaxis; (b) the x - axis 28. Region bounded by y = x3, y = 0 and x = 1 about (a) x = 1 ; (b) y = -1 29. Let R be the region bounded by y = 3-x , the x - axis and the yaxis. Compute the volume of the solid formed by revolving R about the given line. (a) y - axis (b) x - axis (c) y = 3 (d) y = -3 (e) x = 3 (f) x = -3 30. Let R be the region bounded by y = x2 and y = 4. Compute the volume of the solid formed by revolving R about the given line. (a) y = 4 (b) y - axis (c) y = 6 (d) y = -2 31. Let R be the region bounded by y = x2, y = 0 and x = 1. Compute the volume of the solid formed by revolving R about the given line. (a) y - axis (b) x - axis (c) x = 1 (d) y = 1 (e) x = -1 (f) y = -1 32. Let R be the region bounded by y = x, y = -x and x = 1. Compute the volume of the solid formed by revolving R about the given line. (a) x - axis (b) y - axis (c) y = 1 (d) y = -1 33. Let R be the region bounded by y = ax2, y = h and the y - axis (where a and h are positive constants). Compute the volume of the solid formed by revolving this region about the y - axis. Show that your answer equals half the volume of a cylinder of height h and radius . Sketch a picture to illustrate this. 34. Use the result of exercise 33 to immediately write down the volume of the solid formed by revolving the region bounded by and the x - axis about the y - axis. 35. Suppose that the square with -1 x 1 and -1 y 1 is revolved about the y - axis. Show that the volume of the resulting solid is 2. 36. Suppose that the circle x2+y2 = 1 is revolved about the yaxis. Show that the volume of the resulting solid is . 37. Suppose that the triangle with vertices (-1, -1), (0, 1) and (1, -1) is revolved about the y - axis. Show that the volume of the resulting solid is . 38. Sketch the square, circle and triangle of exercises 35-37 on the same axes. Show that the relative volumes of the revolved regions (cylinder, sphere and cone, respectively) are 3:2:1. 39. Verify the formula for the volume of a sphere by revolving the circle x2+y2 = r2 about the y - axis. 40. Verify the formula for the volume of a cone by revolving the line segment about the y - axis. 41. Let A be a right circular cylinder with radius 3 and height 5. Let B be the tilted circular cylinder with radius 3 and height 5. Determine whether A and B enclose the same volume. 42. Determine whether or not the two indicated parallelograms have the same area.