Question

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8. (10 points) Explain in an itemized fashion (preferably with an example) how one can estimate double integrals over non-rectangular domains

Answer 1

Double integrals over rectangular domains we started by considering a cross-section for a fixed value of x. We computed the Area of this cross section using

f(x,y) dy.

น้า Since the volume is the integral with respect to x of the cross-sectional area- A(d We obtained: f(s,y) dA- f(x,y) dy ds Similarly we could start by fixing a value of y

In which case we obtain f (x,y)dxdy f(x.y)dA Suppose however we have a non-rectangular domain, for example suppose we have a circular domain. x, +y's). inthe graph below we have How do we now calculate f(a y) dA The idea is basically the same as with the case of a rectangular domain. We start by fixing a value of x.computing the cross-sectional area by integrating with respect to y and then integrating this result with respect tox

The main difference is that at each value of x the limits of integration change

So let's consider then one cross section 9-(xo Then at 9-(x Since this is true at every x we have in general 9-(x) f (x.y) dy A(X) Therefore since x varies from -3 to 3 9-(x) f(xy)dyd f(x,y) dA A(x) dx= Be careful with double integrals over rectangular regions we could reverse the order of integration without thought. However over non rectangular regions it takes a little more work For example if we were to just ignorantly reverse the order of integration above our result would involve functions of x not a number--volumes and masses are numbers

We will consider a few examples later but let's generalize the 2 types of basic regions we'll consider. Generally we only graph the domain, denoted R, and not the surface z-f(x.y) to set up the double integral. 1. y varies between 2 functions of x and a sxb. P10 f(a,y) dA A dx f(xy) dy d 1. x varies between 2 functions of y and csysd 81C3)

"ค่ไ rd = f(x,y) dA A(y) dy= f(x, y) dxdy

and y #x2 . Suppose f(Xy) #x2 y3 Suppose R is the region bounded by y a. Calculate the volume by first integrating with respect to y then x b. Calculate the volume by first integrating with respect to x then y a. Ateach х, y varies from X. tox. We have y'dy A(x) x varies from O to 1 Therefore we have: y"dyd

dyd: : 013 28 44 28 44 b. Note here we have to invert our functions to get x as a function of y fx.y) dA xy dxdy : 01: 33 21