Paper 3, Section II, C

Vector Calculus | Part IA, 2010

Let f(x,y)f(x, y) be a function of two variables, and RR a region in the xyx y-plane. State the rule for evaluating Rf(x,y)dx dy\int_{R} f(x, y) \mathrm{d} x \mathrm{~d} y as an integral with respect to new variables u(x,y)u(x, y) and v(x,y)v(x, y).

Sketch the region RR in the xyx y-plane defined by

R={(x,y):x2+y22,x2y21,x0,y0}R=\left\{(x, y): x^{2}+y^{2} \leqslant 2, x^{2}-y^{2} \geqslant 1, x \geqslant 0, y \geqslant 0\right\}

Sketch the corresponding region in the uvu v-plane, where

u=x2+y2,v=x2y2u=x^{2}+y^{2}, \quad v=x^{2}-y^{2}

Express the integral

I=R(x5yxy5)exp(4x2y2)dx dyI=\int_{R}\left(x^{5} y-x y^{5}\right) \exp \left(4 x^{2} y^{2}\right) \mathrm{d} x \mathrm{~d} y

as an integral with respect to uu and vv. Hence, or otherwise, calculate II.

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