Volumes of Revolution

Let the region R be the area enclosed by the function f(x)=x^2, the horizontal line y=2, and the y-axis. If the region R is the base of a solid such that each cross section perpendicular to the x-axis is a rectangle whose height is twice the length of its base in the region R, find the volume of the solid. You may use a calculator and round to the nearest thousandth.

• 6.034

• 6.098

• 6.077

• 6.048

Let the region R be the area enclosed by the function

f(x)=√3 and g(x)=3/2x. If the region R is the base of a solid such that each cross section perpendicular to the x-axis is an isosceles right triangle with a base in the region R, find the volume of the solid. You may use a calculator and round to the nearest thousandth.

• 3.35

• 2.89

• 2.40

• 2.65

Let R be the region in the first and second quadrants above by the graph of y=20/(1+x^2) and below by the horizontal line y=2. Find the area of R. You may use a calculator and round to the nearest thousandth.

• 42.344

• 37.980

• 37.961

• 42.386

Let the region R be the area enclosed by the function f(x)=x^2-1, the horizontal line y=-4 and the vertical lines x=0 and x=4. Find the volume of the solid generated when the region R is revolved about the line y=-4. You may use a calculator and round to the nearest thousandth.

• 1159.619

• 1156.532

• 1160.619

• 1158.619

Let R be the region bounded by the graphs of y=sin (πx) and y=x^3-4x. The region R is the base of a solid and for this solid, each cross section perpendicular to the x-axis is a square. Find the volume of this solid.

• 9.978

• 9.950

• 9.560

• 0.576

Let R be the region bounded by the graphs of y=sin (πx) and y=x^3-4x. The region R models the surface of a small pond. At all points in R at a distance x from the y-axis, the depth of the water is given by h(x)=3-x. Find the volume of water in the pond.

• 9.566

• 9.369

• 8.566

• 8.369

Let the region R be the area enclosed by the function f(x)=x^2+2, the horizontal line y=2 and the vertical lines x=0 and x=4. Find the volume of the solid generated when the region R is revolved about the line y=2. You may use a calculator and round to the nearest thousandth.

• 645.576

• 643.398

• 648.988

• 647.342

Let R be the region enclosed by the line y=x and the curve y=1/3x^2. A solid is generated by rotating R about the line y=4. What is the volume of the solid.

• 61π/5

• 29π/5

• 42π/5

• 38π/5

In the figure, R is the shaded region in the first quadrant bounded by the graph of y=4ln(3-x), the horizontal line y=6, and the vertical line x=2. Find the volume of the solid generated when R is revolved about the horizontal line y=8.

• 168.179

• 178.179

• 158.179

• 118.179

In the figure, R is the shaded region in the first quadrant bounded by the graph of y=4ln(3-x), the horizontal line y=6, and the vertical line x=2. The region R is the base of a solid. For this solid, each cross section perpendicular to the x-axis is a square. Find the volume of the solid.

• 18.972

• 31.243

• 22.253

• 26.266