Section 15 9 Change of variables in multiple integrals MATH 251 Fall 2023 From Calculus 1 you learned the substitution rule Z b a f x dx Z d c f g u g0 u du where x g u a g c and b g d We also have substitution rule in double integral and triple integral In fact we have encountered a special case of the rule when we convert a double integral from rectangular coordinates to polar coordinates or when we convert a triple integral from rectangular coordinates to cylindrical or spherical coordinates As an example recall the substitution rule for a double integral from rectangular coordinates to polar coordinates x R f x y dA x S f r cos r sin rdrd Here S is the region in the r plane that corresponds to the region R in the xy plane Note the extra factor r in the formula In general let x g u v and y h u v where g and h are functions of u v that have continuous rst order partial derivatives So for every pair u v you have a corresponding pair x y This is a transformation T a one to one map from the uv plane to the xy plane The below image shows a transformation T that maps a region S in the uv plane to a region R in the xy plane Just like we change from rectangular to polar or from rectangular to spherical there was an extra factor that we had to use in the integrand The extra factor can be found from the Jacobian De nition The Jacobian of the transformation T given by x g u v and y h u v is Change of variables in a double integral Suppose that T is a transformation with nonzero Jacobian and T maps a region S in the uv plane onto a region R in the xy plane Then x y u v x u y u x R f x y dA x S x v y x u y v v f x u v y u v 1 x v y u x y u v dudv 2 Example 1 Find the Jacobian of the transformation x 2u v2 y u v 1 Steps to change variables in double integrals 1 Step 1 compute the Jacobian x y u v 2 Step 2 form the integrand by substituting x and y by their corresponding functions of u and v 3 Step 3 nd the region of integration in terms of u and v 4 Step 4 evaluate the integral Example 2 Use the given transformation to evaluate the following integral sR 3 x2 y2 x y 2dA where R is the rectangle enclosed by x y 0 x y 3 x y 1 and x y 5 u x y v x y Example 3 Use the given transformation to evaluate the following integral sR 6xydA where R is the region bounded by the ellipse 4x2 9y2 36 u 2x v 3y 3 4 Example 4 Use the given transformation to evaluate the following integral sR x y dA where R is the triangular region with vertices 0 0 4 2 and 2 4 x 4u 2v y 2u 4v
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