The formula obtained in part (b) of Exercise 43 is useful in integration problems where it is inconvenient or impossible to solve the transformation equations u = f x , y , υ = g x , y explicitly for x and y in terms of u and υ . In these exercises, use the relationship ∂ x , y ∂ u , υ = 1 ∂ u , υ / ∂ x , y to avoid solving for x and y in terms of u and υ . Use the transformation u = x y , υ = x 2 − y 2 to find ∬ R x 4 − y 4 e x y d A where R is the region in the first quadrant enclosed by the hyperbolas x y = 1 , x y = 3 , x 2 − y 2 = 3 , x 2 − y 2 = 4.
The formula obtained in part (b) of Exercise 43 is useful in integration problems where it is inconvenient or impossible to solve the transformation equations u = f x , y , υ = g x , y explicitly for x and y in terms of u and υ . In these exercises, use the relationship ∂ x , y ∂ u , υ = 1 ∂ u , υ / ∂ x , y to avoid solving for x and y in terms of u and υ . Use the transformation u = x y , υ = x 2 − y 2 to find ∬ R x 4 − y 4 e x y d A where R is the region in the first quadrant enclosed by the hyperbolas x y = 1 , x y = 3 , x 2 − y 2 = 3 , x 2 − y 2 = 4.
The formula obtained in part (b) of Exercise 43 is useful in integration problems where it is inconvenient or impossible to solve the transformation equations
u
=
f
x
,
y
,
υ
=
g
x
,
y
explicitly for x and y in terms of
u
and
υ
.
In these exercises, use the relationship
∂
x
,
y
∂
u
,
υ
=
1
∂
u
,
υ
/
∂
x
,
y
to avoid solving for x and y in terms of
u
and
υ
.
Use the transformation
u
=
x
y
,
υ
=
x
2
−
y
2
to find
∬
R
x
4
−
y
4
e
x
y
d
A
where R is the region in the first quadrant enclosed by the hyperbolas
x
y
=
1
,
x
y
=
3
,
x
2
−
y
2
=
3
,
x
2
−
y
2
=
4.
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.
University Calculus: Early Transcendentals (4th Edition)
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