Part A-

(1) Sketch the region enclosed by the graphs of y = x² and y = 2x + 3 and find the volume of the solid produced when that region is revolved about the x-axis.

(2) Sketch the region enclosed by y = 1/x and y = 5/2 - x, and find the volume of the solid produced when that region is revolved about the y-axis.

(3) Sketch the region in the first quadrant enclosed by y = x³ and y = x^1/3 and find the volume of the solid produced when that region is revolved about the line x = 2.

(4) Use an integral to find the volume of the solid with base the region enclosed by the graphs of y = x⁴ and y = 1 and whose cross sections perpendicular to the y-axis are squares. Draw a picture! You will need to determine the formula for the area of cross-sections to apply the integral formula for volume.

(5) Draw a circle of radius R centered at the origin (x² + y² = R²). Draw a vertical line x = r. (The number r will be positive and less than R so 0 < r < R.) Consider the region to the right of the line and inside the circle. If that region is revolved about the y-axis, bead is formed. Draw a picture! Use an integral to find the volume of the bead.

Part B-

(1) Sketch the region below the graph of y = √x and above the x-axis for 0 ≤ x ≤ 4 and find the volume of the solid produced when that region is revolved about the y-axis.

(2) Sketch the region enclosed by the graphs of y = sin(x²), y = 0, and x = √π, and find the volume of the solid produced when that region is revolved about the y-axis.

(3) Sketch the region enclosed by the graphs of y = x^1/3, y = 3, and x = 0, and find the volume of the solid produced when that region is revolved about the x-axis.

(4) Sketch the region enclosed by the graphs of y = x² and y² = 8x and find the volume of the solid produced when that region is revolved about the line x = 4.

(5) Sketch the region enclosed by the graph of y = 2x - x² and the x-axis find the volume of the solid produced when that region is revolved about the y-axis.