Question

The arc of the parabola y = x^2 from (3,9) to (4,16) is rotated about the y-axis. Find the area of the resulting surface. Please use Woframalpha only to check your final answer. Show all steps in solving the appropriate integral.

Answer 1

Answer:

156.5

Step-by-step explanation:

Thinking process:

The area can be calculated using the formula:

$S = \int\limits^4_3 {2\pi x\sqrt{1+}(2x)^{2} } \, dx$

We let the substitution take place.

Therefore, we let $u = 1 + 4x^{2}$

Thus, $du = 8dx.$

So,

$xdx = \frac{1}{8}du$

Also, the interval of the integration changes to [ 37, 65]

Thus,

$S = \int\limits^4_3 {2\pi \sqrt{1+4x^{2} } } \, dx \\= \int\limits^65_35 {2\pi \sqrt{\frac{1}{8}u } } \, dx$

= $\frac{1}{6} [ 65^{\frac{3}{2}-37^{\frac{3}{2} } }$

= 156.5 units²