Question

Let A = (0, 0), B = (2, 0), and C = (1, 1). Let R be the triangular region in the xy-plane with sides AB, BC, and AC. Set up an integral which gives the volume under the surface f(x, y) = x + y, over the region R and above the xy-plane.

Answer 1

The region
`R` is the set of points

`R=\{(x,y)\mid0\le y\le1,y\le x\le2-y\}`

Then the volume is given by the integral,

`\displaystyle\iint_Rf(x,y)\,\mathrm dA=\int_0^1\int_y^(2-y)(x+y)\,\mathrm dx\,\mathrm dy`