Using the Method of Cylindrical Shells find the volume of the solid generated by revolving about the x-axis the region bounded by the upper half of the ellipse: x^2/a^2 + y^2/b^2 = 1,

Question

and the x-axis and thus find the vol of a prolate spheroid. Here a and b are positive constants, with a>b.

Using the Method of Cylindrical Shells find the volume.

Answer 1

Using cylindrical shells requires that the representative strip in the region to be parallel to the axis of rotation. So, we will have to integrate with respect to y. This means we will want to start by solving for x in terms of y:

x² / a² = (b² - y²) / b²

x = ± a/b √(b² - y²)

We then integrate an expression for the lateral surface area of a cylinder, 2πrh, where r is the vertical distance away from the x-axis and h is the width of each strip (i.e. the width of the region). Thus r = y and h = 2x.

V = ∫₀^b 2πy·2a/b √(b² - y²) dy

= 4aπ/b ∫₀^b [ y · √(b² - y²) ] dy u = b² - y²

du = -2ydy

= -2aπ/b ∫_b^2⁰ [ √u ] du

= -2aπ/b · 2/3u^3/2 ]_b^2⁰

= 4ab²π / 3