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Evaluate $\int_{R} \int f(x, y) d A$ for $R$ and $f$ as given.(a) $f(x, y)=1$(b) $f(x, y)=x y+2 x^{2}+3$

(a) 36(b) $\frac{14148}{5}$

Calculus 3

Chapter 6

An Introduction to Functions of Several Variables

Section 6

Double Integrals

Partial Derivatives

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for this problem we are asked to evaluate the double integral of fx for R and F As given where r is the region in between the lines. Let's see here. It's in between the lines, six X. So that's the top line there and x squared or Y equals six X and Y equals X squared, respectively. So for part A we're just going to be integrating one over that region. So what we can do here, we can see that. It's going to be easiest to first integrate over why then integrate over X. We have a horizontally simple region. So we'd be integrating why from X squared up to six X. And then we're integrating X from one up to six. And we are doing dy dx. The result of that. Innermost integral will be simply six X minus X squared. And we're integrating that from 0 to 6 integral of six X becomes six X squared over two or three X squared integral of X squared becomes X cubed over three. So we have three X squared minus X cubed over three Evaluated from 0 to 6, which gives us a result then of 36. Then for part B we're asked to integrate now I'll integrate in the same order from 0 to 6 over X and from X squared to six X over Y. But we are being asked here to integrate X Y Plus two, x squared plus three. And then we have dy dx. So integrating first over why we'll have the result X times y squared over two plus two, X squared Y plus three Y evaluated from X squared up to six X Squared up to six x dx. So evaluating from endpoint to endpoint we'd have that the integral. And at this step is going to be 18 x -3 x squared Plus 30 x cubed -2X. to the power of four X to the power of four -X to the power of 5/2. And we're integrating that from 0-6, which will then become 18 X squared over two. So that would be nine X squared -3 x cubed over three. So that's just minus x cubed Plus 30 x to the power of 4/4 -2 x. The power 5/5 minus one half or 1/2 times X power of 6/6. So that becomes minus X power of 6/12 Evaluated from 0 to 6. Which will give us the final result result then of 14,148 over five

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