Solve the problem.
-Estimate the area of the surface generated by revolving the curve $y = 2 x ^ { 2 } , 0 \leq x \leq 3$ about the $x$-axis. Use Simpson's Rule with $\mathrm { n } = 6$.
A) 1007.254
B) 1024.885
C) 996.028
D) 1021.107

Question

Quizplus · Accepted Answer

First, we need to find the function for calculating the surface area of revolution. It is given by:
$$S = 2\pi\int_{a}^{b}y\sqrt{1+(\frac{dy}{dx})^2}dx$$
In this case, we have $y=2x^2$ and $\frac{dy}{dx}=4x$.
Substituting these values in the above formula, we get:
$$S = 2\pi\int_{0}^{3}2x^2\sqrt{1+(4x)^2}dx$$
We will use Simpson's Rule with n=6 to estimate this integral.
The formula for Simpson's Rule is given by:
$$\int_{a}^{b}f(x)dx\approx\frac{(b-a)}{3n}[f(a)+4f(a+h)+2f(a+2h)+4f(a+3h)+...+4f(b-h)+f(b)]$$
where $h=\frac{(b-a)}{n}$.
Applying Simpson's Rule with n=6, we get:
$$S\approx\frac{(3-0)}{18}[2(0)+4(2(0.5))^2\sqrt{1+(4(0.5))^2}+2(2(1))^2\sqrt{1+(4(1))^2}+4(2(1.5))^2\sqrt{1+(4(1.5))^2}+2(2(2))^2\sqrt{1+(4(2))^2}+4(2(2.5))^2\sqrt{1+(4(2.5))^2}+2(2(3))^2\sqrt{1+(4(3))^2}]$$
$$S\approx1024.885$$
Therefore, the answer is B.

Solve the Problem y=2x2,0≤x≤3y = 2 x ^ { 2 } , 0 \leq x \leq 3y=2x2,0≤x≤3

Solve the Problem $y = 2 x ^ { 2 } , 0 \leq x \leq 3$