# Calculating Volume using Integrals - Page 5

Questions

- The region under the curve #y=sqrt(1+x^2)# bounded by #0<=x<=1# 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?
- Given #f(x, y)=x^2+y^2-2x#, how do you the volume of the solid bounded by #z=(f(x, y)+f(y,x))/2-5/2, z = +-3?#
- How do you find the volume of the solid in the first octant, which is bounded by the coordinate planes, the cylinder #x^2+y^2=9#, and the plane x+z=9?
- How do you find the volume of the solid generated by revolving the region enclosed by the parabola #y^2=4x# and the line y=x revolved about the x-axis?
- How do you find the volume of the solid generated by revolving the region bounded by the graphs of the equations #y=sec(x)# , y=0, #0 <= x <= pi/3# about the line y = 5?
- How do you find the volume of a solid that is enclosed by #y=3x^2# and y=2x+1 revolved about the x axis?
- The region under the curves #y=x^2, y=x# 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?
- How fast is the volume changing with respect to time when the radius is changing with respect to time when the radius is changing at a rate of dr/dt=1.5 feet per second and r=2 feet?
- How do you find the volume of a solid that is enclosed by #y=sqrt(4+x)#, x=0 and y=0 revolved about the x axis?
- Let R be the region in the first quadrant enclosed by the hyperbola #x^2 -y^2= 9#, the x-axis , the line x=5, how do you find the volume of the solid generated by revolving R about the x-axis?
- The region under the curve #y=1/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 #x=0, x=y-y^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?
- 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 = 5?
- How do you use an integral to find the volume of a solid torus?
- How do you find the volume of the solid with base region bounded by the curves #y=1-x^2# and #y=x^2-9# if cross sections perpendicular to the #x#-axis are squares?
- Let R be the shaded region in the first quadrant enclosed by the y-axis and the graphs of #y=4-x^2# and #y=1+2sinx#, how do you find the volume of the solid whose base is R and whose cross sections perpendicular to the x-axis are squares?
- How do you find the volume of the solid bounded by x^2+y^2=4 and z=x+y in the first octant?
- The region under the curves #y=x^3, y=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?
- What is the volume of the solid given the base of a solid is the region in the first quadrant bounded by the graph of #y=-x^2+5x-4^ and the x-axis and the cross-sections of the solid perpendicular to the x-axis are equilateral triangles?
- How do you find the volume of the solid with base region bounded by the curve #9x^2+4y^2=36# if cross sections perpendicular to the #x#-axis are isosceles right triangles with hypotenuse on the base?