Volume Calculation Using the Shell Method

The video tutorial explains how to find the volume of a solid formed by rotating a region in the first quadrant around the y-axis. The shell method is used, involving setting up and evaluating an integral. The process includes identifying the function, axis of rotation, and setting up the integral with p(x) and h(x). The integral is evaluated to find the exact and approximate volume of the solid, with a final result of 8π cubic units or approximately 25.1 cubic units.

What is the region being rotated to find the volume of the solid?

• The region bounded by y = x^3, y = 0, and x = 3

• The region bounded by y = x, y = 0, and x = 1

• The region bounded by y = x^2, y = 1, and x = 2

• The region bounded by y = x^2, y = 0, and x = 2

What is the axis of rotation in this problem?

• z-axis

• Line x = 2

• x-axis

• y-axis

Which method is used to find the volume of the solid?

• Shell Method

• Washer Method

• Disk Method

• Cavalieri's Principle

Why is the shell method appropriate for this problem?

• Because the region is bounded by a circle

• Because the function is a function of x and the axis of rotation is vertical

• Because the axis of rotation is horizontal

• Because the function is a function of y

What does p(x) represent in the shell method?

• The thickness of the shell

• The height of the shell

• The radius of the disk

• The distance from the axis of rotation

What is the expression for h(x) in this problem?

• h(x) = 2x

• h(x) = x^3

• h(x) = x

• h(x) = x^2

What is the integrand in the integral used to find the volume?

• x^2

• x^3

• 2x^2

• x^4

What is the antiderivative of x^3?

• x^2 / 2

• x^5 / 5

• x^3 / 3

• x^4 / 4

What is the exact volume of the solid in cubic units?

• 4π

• 2π

• 8π

• 16π

What is the approximate volume of the solid in cubic units?

• 50.2

• 25.1

• 31.4

• 12.5