We saw with polar coordinates we may be able to simplify a integral of the form [ ],f(x, y, z) dA by making a change of variables (that is, a substitution) of the form T :x=r Cos(8), and y=r Sin(@). The equations x= rCos(8), and y = rSin(@) convert , and 8 to x, and y and maps a region Pin the r 0- space to a region P' in the xy-space; we call these formulas the change of variable formulas. For example the consider the polar rectangle P- [1, 2] x [1, 2] in the /0-space. If we use change of variable formulas to transform every point in P then we'll get a new shape P' inxy-space. Does that shape look familiar? T 25 20 T(2.1) 1.5 (1.1) (2.1) RT ( 1, 1) P P T (2,.5) (2,5) T( 1,.5) (1..5) X 0.0 15 2.0 25 10 10.0 19 20 25 3.0 We can see that if we wanted to find the volume of a surface over region P' by using the region P we would need some scaling factor, because P is bigger than P. To find that factor we see how tiny rectan- gles in r 0-space look in x y-space and compare their area. Consider the rectangle P=[u, v] x [u + Au, v+ Av] in the u v-space. When we transform this region into the xy-space we get the region P'y T(U+Au,v+AV) (U.V+AV) (U+AU,V+AV) T(U.V+AV) P P T(U+Au,V) (u,V) (U+AU.V) U T(U.V) X Although the sides of P' aren't necessarily straight. we will approximate the area as the area of a parallel ogram whose sides are given by the vectors from T(v. V) toT (u + Au, v) and T(u. v) to T(u, v + Av). Let's call those vectors i and w respectively. The components of v are ir = (x(U + Au, v) - x(u, v). y(u + Au, v) - y(u, v)> and similarly those for ware w = (x(u, v + Av) - x(u, v). y(u, v + Av) - y(u, vi> y T(U+Au,V+Av) P T(U.V+AV) T(u+Au,v) T(u,v) X We can approximate i asAu and find a similar approximation for wv. Find the volume of the parallelogram spanned by the vectors ("s, ") Au, and the approximation you get for w and write a report, two page maximum, that convinces an audience of your peers that [JA(x. y) dxdy = [[_f((u, v). yu. v)) | = =- dudv. If you find it helpful, choose a specific example to present to your audience. Guide Below is a guide to help you explore the situation. It's not necessary that you follow the guide but it is a great place to start and every report should address the questions in the guide, however your report should read like a single coherent document and not just answers to the guided questions below. 1. What's the similar approximation for w? 2. What's relationship between the area of the parallelogram you find and the area of P? 3. Why do you care about their relationship