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

156.5

Explanation:

Thinking process:

The area can be calculated using the formula:

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

We let the substitution take place.

Therefore, we let
`u = 1 + 4x^(2)`

Thus,
`du = 8dx.`

So,

`xdx = (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{(1)/(8)u } } \, dx`

=
`(1)/(6) [ 65^{(3)/(2)-37^{(3)/(2) } }`

= 156.5 units²