Area by Integration
Find the area under the function y=x3 from x=1 to x=2.
15/4
Write an integral to represent the volume generated when the area in the first quadrant bounded by y=x2 and the line x=4 is rotated about the x-axis.
π∫04 [x2]2dx
Write an integral to represent the volume of the solid generated when the area between y=x and y=x2 is rotated about the x=axis.
π∫01 [(x)2-(x2)2]dx
The region R in the first quadrant between y=∛x and x=2 is the base of a solid. For this solid, the cross sections perpendicular to the x-axis are rectangles whose height is six times the length of the base. Write an integral to represent the volume of the solid.
6∫02(∛x)2dx
Find the area of the region bounded by y=x and y=x2
1/6
What is the volume of the solid generated when the area in the first quadrant bounded by y=x and the line x=3 is rotated about the x-axis?
9π
Write an integral to represent the volume of the solid generated when the area between y=√(x) and y=x2 is rotated about the line y=-2.
π∫01[(√(x)-(-2))2-(x2-(-2))2]dx or ∫01[(√(x)+2)2-(x2+2)2]dx
The region R between y=4 and y=x2 is the base of a solid. For this solid, the cross sections perpendicular to the x-axis are squares. Write an integral to represent the volume of the solid.
∫-22(4 - x2)2dx
Find the area of the region bounded by y=√x and y=x/4
32/3
Write an integral to represent the volume generated when the area in the first quadrant bounded by y=x2 and the line y=4 is rotated about the y-axis.
π∫04 [√y]2dy
Write an integral to represent the volume of the solid generated when the area between y=√(x) and y=x2 is rotated about the line y=5.
π∫01[(5 - x2)2 - (5 - √(x))2]dx
The region R between y=2x and y=x2 is the base of a solid. For this solid, the cross sections perpendicular to the y-axis are semicircles. Write an integral to represent the volume of the solid.
π/8∫04(√y - y/2)2dy
Find the area bounded by the functions y=x2-3 and y=2x.
32/3
Determine the volume of the solid generated when the area bounded by y=√x and y=2 is rotated about the y-axis.
32π/5
Write an integral to represent the volume of the solid generated when the area between y=√x and y=x/3 is rotated about the y-axis.
π∫03 [(3y)2-(y2)2]dy
The region R between y=x2 and y=x is the base of a solid. For this solid, the cross sections perpendicular to the x-axis are squares. Find the volume of the solid.
∫01(x - x2)2dx
∫01(x2 - 2x3 + x4)dx
Volume: 1/30