Find the Jacobians ∂ x , y / ∂ u , v of the given transformations from variables x , y to variables u , v In the integral I = ∫ x = 0 1 / 2 ∫ y = x 1 − x x − y x + y 2 d y d x , make the change of variables x = 1 2 r − s , y = 1 2 r + s , and evaluate I. Hints: See Problem 19. To find the r and s limits, sketch the area of integration in the (x,y) plane and sketch the r and s axes. Then show that to cover the same integration area, you may take the r and s limits to be: s from 0 to r, r from 0 to 1.
Find the Jacobians ∂ x , y / ∂ u , v of the given transformations from variables x , y to variables u , v In the integral I = ∫ x = 0 1 / 2 ∫ y = x 1 − x x − y x + y 2 d y d x , make the change of variables x = 1 2 r − s , y = 1 2 r + s , and evaluate I. Hints: See Problem 19. To find the r and s limits, sketch the area of integration in the (x,y) plane and sketch the r and s axes. Then show that to cover the same integration area, you may take the r and s limits to be: s from 0 to r, r from 0 to 1.
Find the Jacobians
∂
x
,
y
/
∂
u
,
v
of the given transformations from variables
x
,
y
to variables
u
,
v
In the integral
I
=
∫
x
=
0
1
/
2
∫
y
=
x
1
−
x
x
−
y
x
+
y
2
d
y
d
x
,
make the change of variables
x
=
1
2
r
−
s
,
y
=
1
2
r
+
s
,
and evaluate I. Hints: See Problem 19. To find the r and s limits, sketch the area of integration in the (x,y) plane and sketch the r and s axes. Then show that to cover the same integration area, you may take the r and s limits to be: s from 0 to r, r from 0 to 1.
With differentiation, one of the major concepts of calculus. Integration involves the calculation of an integral, which is useful to find many quantities such as areas, volumes, and displacement.
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