# Calculating Volume using Integrals - Page 6

Questions

- The region under the curve #y=x# bounded by #1<=x<=2# is rotated about a) the x axis and b) the y axis. How do you sketch the region and find the volumes of the two solids of revolution?
- The region under the curves #y=cosxsqrtsinx, 0<=x<=pi/2# is rotated about the x axis. How do you sketch the region and find the volumes of the two solids of revolution?
- The region under the curves #y=xe^(x^3), 1<=x<=2# is rotated about the x axis. How do you find the volumes of the two solids of revolution?
- How do you find the volume of the pyramid bounded by the plane 2x+3y+z=6 and the coordinate plane?
- How do you use Lagrange multipliers to find the volume of the largest rectangular box in the first octant with three faces in the coordinate planes and one vertex in the given plane x + 8y + 5z = 24?
- The first quadrant region enclosed by y=2x, the x-axis and the line x=1 is resolved about the line y=0. How do you find the resulting volume?
- Let R be the shaded region in the first quadrant enclosed by the y-axis and the graphs of #y=sin(x)# and #y=cos(x)#, how do you find the volume of the solid whose base is R and whose cross sections, cut by planes perpendicular to the x-axis, are squares?
- How do you find the volume of the largest rectangular box in the first octant with three faces in the coordinate planes and one vertex in the plane x + 2y + 3z = 4?
- Given #{r,s,u,v} in RR^4# Prove that #min {r-s^2,s-u^2,u-v^2,v-r^2} le 1/4#?
- How do you find the volume of a solid y=x^2 and x=y^2 about the axis x=–8?
- Let R be the region in the first and second quadrants bounded above by the graph of #y=20/(1+x^2)# and below by the horizontal line y=2, how do you find volume of the solid generated when R is rotated about the x-axis?
- How do you find the volume of the solid whose base is the region bounded by y = x^2, y =x, x = 2 and x = 3, where cross-sections perpendicular to the x-axis are squares?
- What is the volume of the solid generated when S is revolved about the line #y=3# where S is the region enclosed by the graphs of #y=2x# and #y=2x^2# and x is between [0,1]?
- How do you use a triple integral to find the volume of the given the tetrahedron enclosed by the coordinate planes 2x+y+z=3?
- The base of a solid region in the first quadrant is bounded by the x-axis,y-axis, the graph of #y=x^2+1#, and the vertical line x=2. If the cross sections perpendicular to the x-axis are squares, what is the volume of the solid?
- The region under the curves #y=sqrt(e^x+1), 0<=x<=3# is rotated about the x axis. How do you sketch the region and find the volumes of the two solids of revolution?
- How do you find the volume of the central part of the unit sphere that is bounded by the planes #x=+-1/5, y=+-1/5 and z=+-1/5#?
- Let R be the shaded region in the first quadrant enclosed by the y-axis and the graphs of #y=sin(x)# and #y=cos(x)#, how do you find the volume of the solid generated when R is revolved about the x-axis?
- Let R be the region in the first quadrant bounded by the graph of #y=8-x^(3/2)#, the x-axis, and the y-axis. What is the best approximation of the volume of the solid generated when R is revolved about the x-axis?
- The region under the curve #y=sqrt(2x-4)# bounded by #2<=x<=4# is rotated about a) the x axis and b) the y axis. How do you sketch the region and find the volumes of the two solids of revolution?