Double integration - finding the limits

[PhyStan7]

Hello, I am stuck on this double integration question as I am not sure how to get the limits to integrate between. If anyone could give me advice on ways to get limits in general, help would be appreciated.

Homework Statement

Evaluate the following integral

_R(xy+cosx)dxdy

where R, the region of integration, is the triangle with vertices at the points

(x,y)=(0,0),(2,0),(1,1)

The Attempt at a Solution

Ok so after drawing a diagram it is clear, taking y as the inner integral, that 0<x<2 are the limits to take for x. It can be seen that y must be above 0 so i thought that would be the lower limit but have no idea what the upper limit of y would be. I know that the sides of the shape are y=x and y=-x+2. Would i have to take the triangle as 2 different shapes, where for shape 1 0<y<y=x and shape 2 0<y<y=-x+2 and then add the shapes?

Help would be appreciated, thanks! :)

 

[Mark44]

If you integrate with respect to y first you will need two iterated integrals, corresponding to the two triangular regions than make up the larger triangular region. In the first iterated integral, y ranges between y = 0 and y = x; in the second integral, y ranges between y = 0 and y = -x + 2. The limits for integration in both integrals is x = 0 to x = 2.

If you integrate with respect to x first, you don't need two iterated integrals. The limits on the inner integral are x = y to x = -y + 2. The limits on the outer integral are y = 0 to y = 1.

Hope that helps.