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**Question**:

A solid has a rectangular base that lies in the first quadrant and is bounded by the x and y-axes and the lines x=2, y=1. The height of the solid above point (x,y) is 1+3x. Find the Riemann approximation of the solid.

**Solution**:

I already know that the solution is \(\displaystyle \sum_{i=1}^{n} \frac{2}{n} \left(1+\frac{6i}{n} \right)\). What I don't see is why it's 1+(6i)/n and not 1+(3i)/n. Volume can be generalized to be the area of the base times the height, so for this problem I have something like \(\displaystyle x*y*f(x)\). Of course x is changing so I must rewrite this.

For any partition where we approximate the volume between x_1 and x_2 the length will be \(\displaystyle \Delta x=\frac{2}{n}\) The y value is a constant 1, so won't need to be written explicitly as far as I can see. I know this part is incorrect but it seems to me that the height should be \(\displaystyle 1+\frac{3i}{n}\), but I know that since we haven't defined where \(\displaystyle x_i\) is in each partition (it could be the left value, middle value, right value or anywhere) then I'm really stuck here.

EDIT: Now that I think about it more since we haven't defined what n is we don't know what i/n either and i/n will change according to how many partitions we take. So am I correct in thinking that \(\displaystyle f(x_i)=f(i\Delta x)\)? If so this makes sense now.

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