Hello,(adsbygoogle = window.adsbygoogle || []).push({});

I was helping my friend prepare for a calculus exam today - more or less acting as a tutor.

He had the following question on his exam review:

∫∫_{R}y^{2}dA

Where R is bounded by the lines x = 2, y = 2x + 4, y = -x - 2

I explained to him that R is a triangle formed by all three of those lines. The solution to the integral is 192, which is pretty easy to calculate by hand without a calculator.

He asked me why the solution wasn't just the area of R (two triangles if you split R at the line y = 0) which is 128. I told him its because we are integrating y^{2}with respect to the given boundaries, and that if we were integrating "1" it would be the area of the region R. However, we then proceeded to integrate "1" with the same boundaries, and the answer was 24, not 128.

So I ended up coming up with my own question. I can draw a pretty good geometric interpretation of most integration problems including three dimensional ones given in different coordinate systems... but I feel like I'm missing something primitive here. Can anyone help me develop a geometric picture of what this type of double integral is representing?

Thanks!

**Physics Forums | Science Articles, Homework Help, Discussion**

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Graphical interpretation of a double integral?

**Physics Forums | Science Articles, Homework Help, Discussion**