Double integrals—your choice of transformation Evaluate the following integrals using a change of variables. Sketch the original and new regions of integration , R and S. 32. ∬ R y 2 − x 2 d A , where R is the diamond bounded by y – x = 0, y – x = 2, y + x = 0, and y + x = 2
Double integrals—your choice of transformation Evaluate the following integrals using a change of variables. Sketch the original and new regions of integration , R and S. 32. ∬ R y 2 − x 2 d A , where R is the diamond bounded by y – x = 0, y – x = 2, y + x = 0, and y + x = 2
Solution Summary: The author evaluates the value of the integral and sketches the original and new region.
Double integrals—your choice of transformationEvaluate the following integrals using a change of variables. Sketch the original and new regions of integration, R and S.
32.
∬
R
y
2
−
x
2
d
A
, where R is the diamond bounded by y – x = 0, y – x = 2, y + x = 0, and y + x = 2
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.
Consider the following.
y
y = x+2
y = x?
-3
-2
-1
1
(a) Find the points of intersection of the curves.
(х, у) -
(smaller x-value)
(х, у) %3D
(larger x-value)
(b) Form the integral that represents the area of the shaded region.
dx
(c) Find the area of the shaded region.
3.
Sketch the region of integration and change the order of integration.
3 √9 - y²
Br
-3 JO
f(x, y) dx dy
f(x, y) dy dx
The graph of g consists of two straight lines and a semicircle as shown in the figure.
y
20
y = g(x)
10
20
Evaluate each integral by interpreting it in terms of areas.
9(x) dx
(b)
9(x) dx
(c) 9(x) dx
University Calculus: Early Transcendentals (4th Edition)
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