calculus question

Question

set up integrals that can be used to find the volume of solid generated if the region bounded by y=9-x^2 and the line y=5 is rotated about the axis indicated in parts (a) the x-axis and part (b) the line y=-3

Answer 1

Using the washer method (π ∫_a^b [outer-radius² - inner-radius²]dx means it is necessary to determine expressions for the outer and inner radii for each of the two axes of revolution.

About x-axis

Outer Radius for a given x value is the distance from y = 0 to corresponding point on 9 - x². Think y-coordinate at the top minus y-coordinate at the bottom for a vertical segment. O.R. = (9 - x²) - 0; O.R.² = 81 -18x² + x⁴.

Inner radius for a given x-value is a constant: distance from x-axis (y= 0) to y = 5. I.R. = 5 - 0 =5. I.R.² = 25

Since the parabola is symmetric about the y-axis the integral can be twice the integral from x = 0 to x = 2. Note that when y = 5, x = ±2.

2 π ∫₀² [(x⁴ -18x² + 81) - 25]dx

About y = -3

This time the outer radius is (9 - x²) - (-3) = 12 - x². O.R.² = 144 - 24x² + x⁴.

Inner radius is 5 - (-3) = 8. I.R.² = 64

2 π ∫₀² [(x⁴ - 24x² + 144) - 64]dx

This graph shows the results for evaluating the definite integrals:

desmos.com/calculator/dthxtfbeg9