We're currently doing integration on areas between two or more functions, and i'm having trouble with it.
These are the instructions for most of the questions in the homework:
"Sketch the region enclosed by the given curves. Decide whether to integrate with respect to x or y. Find the area of the region."
I was wondering if you could explain the general procedure behind this? Here are two of the problems I attempted, but which I don't think are correct.
1) y = sinx, y = x, x = pi/2, x = pi
I did: integral from pi/2 to pi of: (x - sinx)dx
= (1/2)*x^2 + cosx | pi/2 to pi
= [pi^2/2 - 1] - [pi^2/8 + 0] = -pi^2/2 - 1 = -5.9348.
Negative? I didn't think that could be right. I'm also kind of confused about which of those functions is considered to be the 'top' , since the area between the function is the integral of f(x)-g(x). On the graph it looks like they touch each other.
2) x = y^2 - 4y, x = 2y - y^2. from y = 0 to y = 3
I did: integral from 0 to 3 of (y^2 - 4y - 2y + y^2)dy
= integral from 0 to 3 of (2y^2 - 6y)dy
= (2/3)*y^3 - 3y^2 | 0 to 3
= [18 - 27] - [0 - 0] = -9? Another negative