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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) $f(x, y)=2 x^{2}+3 y^{2}+4$

(a) $1 / 3$(b) $37 / 21$

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 find the double integral of F across the region are which I'll note is bounded by the curves below Y equals x squared and above Y equals root X. Where for part A we have F of X Y is just equal to one. So we're just finding the area of our region. Can see that we have a horizontally simple region. So let's integrate over why first. So we're integrating from X squared up to X. And then we're integrating X from we can see it's 0-1 of dy dx which will then become the integral from 0 to 1 of root X minus x squared dx Which will give us a result then of 1/3. Or excuse me, I'm getting ahead of myself integrating Root X would give us 2/3 times X. The power of 3/2 integrating X squared gives us x cubed over three. We're evaluating this from 0-1 which gives us our final result of 1/3. Then for part B we're asked to integrate over the same region but this time with f of x equals two, X squared plus three, Y squared plus four. So we'll integrate in the same order here, X squared up to root X. Inside. So our first integrating over why we have two X squared plus three Y squared plus four dy dx. So first integrating over X we'd have two X to the power of 3/3. Alright, excuse me we're integrating over Y. First so that becomes two X squared Y plus three Y. To the power of 3/3. Or just Y to the power of three plus four? Y. Evaluated from X squared to root X dx evaluating that. Inner into grand From endpoint to endpoint. Well give us let's see here, I'll unfold everything so it's a little bit easier to work with. We have four X. The power of one half Plus X to the power of 3/2 -4 x squared Plus two x to the power of 5/2 minus two x. The power four Plus X to the power of six DX. Then integrating this over X. We'd have four times 2/3 Times x power of 3/2 plus uh to over five X to the power of 5/2 -4 x cubed over three Plus two times. Let's see here would be two times to over seven times x power of 7/2 -2 x. to the power of 5/5 Plus X. The power of 7/7, evaluated from 0-1. Which will give us the final result here of 20. or excuse me, it's 37 over 21

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