The transformation x=au, y=bv (a>0, b>0) can be rewritten as x/a=u, y/b=v, and hence it maps the curricular region u2+v2≤1 into the elliptical region x2/a2+y2/b2≤1. In these exercises, perform the integration by transforming the elliptical region of integration into a circular region of integration and then evaluating the transformed integral in polar coordinates. ∬R16x2+9y2dA

,where R is the region enclosed by the ellipse

(x2/9)+(y2/16)=1

96π

Step-by-step explanation:

The transformation x = au , y = bv (a>0, b>0) can be re-written as;

which maps the circular region u² + v² ≤ 1 into the elliptical region

Integrating and transforming the elliptical region into a circular region;

here;

R = region enclosed by the ellipse.

Let x = 3u ; y = 4v

Jocobian = J(x,y)

So;

Using polar coordinates;

∴

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