CALCULUS AB VOLUME – REVIEW (Disk Method, Washer Method, Shells Method, Cross Sections)
SHOW ALL WORK IN YOUR NOTEBOOK
The base region for all the following solids with known cross sections is the circle with equation: xy229 . Find the volume in each case if the cross sections are parallel to the y-axis (perpendicular to the x-axis).
1. Cross Section: Circles, diameter in the xy=plane 2. Cross Section: Squares, side in the xy-plane 3. Cross Section: Equilateral Triangles, side in the xy-plane 4. Cross Section: Squares, the diagonals in the xy-plane 5. Cross Section: Isosceles Right Triangles, one of the legs in the xy-plane 6. Cross Section: Semicircles, diameter in the xy-plane
7. Find the volume of the solid that lies between planes perpendicular to the x- axis at x = -1 and x = 1. The cross sections perpendicular to the x-axis between these planes are squares whose diagonals run from the semicircle yx1 2 to the semicircle yx 1 2 .
8. The base of a solid is the region between the curve yx 2sin() and the interval [0, ] on the x-axis. Find the volume if the cross sections perpendicular to the x-axis are
a) equilateral triangles with bases running from the x-axis to the curve. b) Squares with bases running from the x-axis to the curve.
9. Find the volume of the solid generated by revolving the region enclosed by the parabola yx2 4 and the line y = x about
a) the x-axis b) the y-axis c) the line x = 4 d) the line y = 4
10. Find the volume of the solid generated by revolving the region bounded by the x-axis, the curve yx 3 4 , and the lines x = -1 and x = 1 about the x-axis.
x2 11. The section of the parabola y from y = 0 to y = 2 is revolved about the 2 y-axis to form a bowl.
a) Find the volume of the bowl. b) Find how much the bowl is holding when is filled to a depth of k units (0 12. Let R be the region in the first quadrant enclosed by the y-axis and the graphs of yx2sin() and yx sec( ) . a) Find the area of R. b) Find the volume of the solid generated when R is revolved about the x-axis. c) Find the volume of the solid whose base is R and whose cross sections cut by planes perpendicular to the x-axis are squares. MULTIPLE CHOICE SECTION OF REVIEW 1. 2. 3. 4. 5.