Verified answer

The bounded volume is the volume above the rectangle in the x-y plane bounded by the lines x = 0, x = 2, y = 0 and y = 1 and below the plane x + y + z = 3. This latter equation can also be written

z = 3 - x - y.

Thus an integral which computes the bounded volume is

integral (x = 0 ---> 2) integral (y = 0 ---> 1) (3 - x - y) dy dx

= integral (x = 0 ---> 2) [(3 - x)y - y^2/2] | (y = 0 ---> 1) dx

= integral (x = 0 ---> 2) [(3 - x) - 1/2] dx

= integral (x = 0 ---> 2) (5/2 - x) dx

= (5x/2 - x^2/2) | (x = 0 ---> 2)

= 5 - 2

= 3

Of course you don't need to use calculus to find the bounded volume, but the question specifies that calculus is to be used.