How to compute the volume of the solid bounded by the graph z=x2+yz=x^{2}+y, the rectangle R=[0,1]×[1,2]R=[0,1]\times [1,2] and the “vertical sides” of RR?

Problem:

Compute the volume of the solid bounded by the graph z=x2+yz=x^{2}+y, the rectangle R=[0,1]×[1,2]R=[0,1]\times [1,2] and the “vertical sides” of RR.

Solution:

I know that z=x2+yz=x^{2}+y is a parabolic cylinder, but I don’t know how to draw that rectangle RR in 3D.

Some ideas, suggestions,…?

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1 Answer
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When the region RR is bounded by curves it’s nice to have a graph of it so you can tell what the bounds will be. But even then you don’t need a 3D graph, it’s a region in the xyxy plane.

But when the region is a rectangle, as in this case, it’s even easier. The sides of the rectangle are your bounds. No need to graph anything, Just evaluate

∫21∫10×2+ydxdy=∫211/3+ydy=11/6\int_1^2 \int_0^1 x^2+y \, dx\,dy
= \int_1^2 1/3+y\,dy
=11/6

If you do want to visualize the graph, just picture that portion of the quadriceps surface over the rectangle in question.