Question

Define the surface area of revolution of the surface generated by resolving the graph of y = f(x) from x = a to x = b about the x axis.

Answer 1

Start with the point (x*, f(x*)). Revolving this point about the x-axis gives you a circle with radius f(x*), so its circumference is
\[C_{x^*}=2\pi f(x^*)\]
Now think of every point on the curve f(x). Each point, upon revolution, forms a circle with radius f(x) for each given x.

The area of the surface of revolution is basically the sum of all these circumferences. As an integral, it's
\[A=\int_a^b 2\pi f(x)\;dx\]

Answer 2

Thank You

Answer 3

I skipped a bunch of details, but that was a somewhat informal explanation of how to find the area of a surface of revolution.