Question

To find the area of the surface obtained by rotating the curve 3x=y+y3 about the x-axis and the y-axis, you can set up double integrals for each case. For rotation about the x-axis:

• The surface area is given Ax=2π∫abf(y)1+f′(y)2dy
And for rotation about the y-axis:
• The surface area is given by Ay=2π∫cdx1+(dydx)2dy

Answer 1

Calculating the surface area created by rotating a curve around an axis involves setting up and evaluating surface integrals, where you take antiderivatives of both dimensions that define the area.

Step-by-step explanation:

The question concerns calculating the surface area of a shape formed by rotating a curve around an axis, which is a problem in multivariable calculus. The surface area for the rotation about the x-axis is given by Ax = 2π∫abf(y)√1+f'(y)^2dy, and about the y-axis by Ay = 2π∫cdx√1+(dy/dx)^2dy. These are formulas involving surface integrals, where integration is used to accumulate infinitesimal area elements over a surface in space. When dealing with surface integrals, the approach is to take antiderivatives of both dimensions that define the area, and the bounds of the integral are determined by the edges of the surface. As such, calculating the surface area after rotating a curve involves setting up and evaluating these integrals appropriately.