For each exercise, write a definite integral, with respect to the indicated variable, that expresses the area or volume being described. Then, evaluate that definite integral using the fundamental theorem of calculus.
Note: The values of each pair of definite integrals (displayed in the same color) should be equal!
1) Area of R1 (Integrate with respect to x).
2) Area of R1 (Integrate with respect to y).
3) Area of R2 (Integrate with respect to x).
4) Area of R2 (Integrate with respect to y).
5) Area of R3 (Integrate with respect to x).
6) Area of R3 (Integrate with respect to y).
7) Volume of solid of revolution formed by rotating R1 about the x-axis
(Integrate with respect to x.)
8) Volume of solid of revolution formed by rotating R1 about the x-axis
(Integrate with respect to y.)
9) Volume of solid of revolution formed by rotating R1 about the y-axis
(Integrate with respect to x.)
10) Volume of solid of revolution formed by rotating R1 about the y-axis
(Integrate with respect to y.)
11) Volume of solid of revolution formed by rotating R2 about the x-axis
(Integrate with respect to x.)
12) Volume of solid of revolution formed by rotating R2 about the x-axis
(Integrate with respect to y.)
13) Volume of solid of revolution formed by rotating R2 about the y-axis
(Integrate with respect to x.)
14) Volume of solid of revolution formed by rotating R2 about the y-axis
(Integrate with respect to y.)
15) Volume of solid of revolution formed by rotating R3 about the x-axis
(Integrate with respect to x.)
16) Volume of solid of revolution formed by rotating R3 about the x-axis
(Integrate with respect to y.)
17) Volume of solid of revolution formed by rotating R3 about the y-axis
(Integrate with respect to x.)
18) Volume of solid of revolution formed by rotating R3 about the y-axis
(Integrate with respect to y.)