Calculus 3

Question

find the volume of the region bounded above by the plane z=y/2 and below  by the rectangle R:0≤x≤4, 0≤y≤2.

Answer 1

The volume described is a triangular prism. Along the y-axis, the height is 0, linearly increasing to 1 when y=2. Using geometric techniques, you can find that the area will be 1/2(1)(2)(4) = 4. Now lets see if we find get that using calculus.

 

If you slice the volume along the x and y axes, the slices will have width dy, height y/2 and length dx.

 

∫_y=0² ∫_x=0 ⁴ y/2 dx dy

 

The height is constant over x, so ∫₀⁴ dx = 4 and you get:

 

∫_y=0² 2y dy

 

Which is an integral you should be able to do in your sleep by now. Did you get 4? Good!