Question

Use Simpson’s Rule with n = 10 to approximate the area of the surface obtained by rotating the curve about the x-axis. Compare your answer with the value of the integral produced by your calculator.

Answer 1

The answer is "7.248934".

Explanation:

The area of the curve obtained after rotating it about the x-axis is :

`2 \pi \int^2_1 y \sqrt{1+ \{ (dy)/(dx)\}^2 \ dx}\\\\y=x\ \ln \ x \ And \ (dy)/(dx)=1+ \lh\ x`

So, The area of the curve obtained after rotating it about the x-axis is :
`2 \pi \int^2_1 (x \ln \ x) √((\ln\ x)^2+ 2 \ln \ x+ 2\ dx)\\\\`

Simpson's rule approximation with n=10 is:

`((1)/(3))* (0.1) * ( f(1) + 4 * f(1.1) + 2* f(1.2) + 4 * f(1.3) + 2 * f(1.4) + 4 * f(1.5) + 2 * f(1.6) + 4 * f(1.7) + 2 * f(1.8) + 4 * f(1.9) + f(2) ) = 7.248933= 7.248934`