# How do you find the area of the surface generated by rotating the curve about the y-axis x=2t+1, y=4-t, 0<=t<=4?

Jul 17, 2017

First we will combine this 2 equation to find x in term of y and then we will calculate the area.

#### Explanation:

$y = 4 - t \iff t = 4 - y$
we plug this value into $x = 2 t + 1 \iff x = 2 \left(4 - y\right) + 1 \iff x = 9 - 2 y$
this gives us the curve
graph{x=9-2y [-10, 10, -5, 5]}
and $0 \le y \le 4$
the area generated is given by the integral:
$E = {\int}_{0}^{4} \left(9 - 2 y\right) \mathrm{dy}$
because we rotate the curve about the y-axis
so $E = {\left[9 y - {y}^{2}\right]}_{0}^{4} = 36 - 16 = 20$