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

Answer:

The answer is "7.248934".

Step-by-step explanation:

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

$2 \pi \int^2_1 y \sqrt{1+ \{ \frac{dy}{dx}\}^2 \ dx}\\\\y=x\ \ln \ x \ And \ \frac{dy}{dx}=1+ \lh\ x$

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

Simpson's rule approximation with n=10 is:

$(\frac{1}{3})\times (0.1) \times ( f(1) + 4 \times f(1.1) + 2\times f(1.2) + 4 \times f(1.3) + 2 \times f(1.4) + 4 \times f(1.5) + 2 \times f(1.6) + 4 \times f(1.7) + 2 \times f(1.8) + 4 \times f(1.9) + f(2) ) = 7.248933= 7.248934$