Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.
a. Assuming g is integrable and a, b, c. and d are constants,
b.
c. Changing the order of
d. The transformation T: x = v, y –u maps a square in the uv-plane into a triangle in the xy-plane.
a.
Answer to Problem 1RE
The statement is not true.
Explanation of Solution
Given:
The integrable function g and the constants a,b,c,d.
Theorem used:
Fubini’s Theorem: Let f be continuous on the rectangular region
The double integral of f over R may be evaluated by either of two iterated integrals:
Reason:
Use the above mentioned Fubini’s theorem to prove this statement.
When we write
Consider the example the volume of a solid bounded by the surface
On further simplification,
Implify the Right hand side of the give equation,
On further simplification
From the equations (1) and (2), we can find that the evaluated values are not the same.
Hence, the statement is not true.
b.
Answer to Problem 1RE
The statement is true.
Explanation of Solution
Given:
Reason:
The set
Thereby the set is same as the remaining two sets
Hence, the statement is true.
c.
To find: Whether the change of order of integration in
Answer to Problem 1RE
The statement is not true.
Explanation of Solution
Given:
The integral
Theorem used:
Let f be continuous over the region
where g,h,G and H are continuous functions. Then f is integrable over D and the triple integral is evaluated as the iterated integral
Reason:
Consider the example,
Use the above theorem to change the order of integration in the above example,
It is observed that the change in order of integration does not alter the integrand.
Hence, the statement is not true.
d.
Answer to Problem 1RE
The statement is not true.
Explanation of Solution
Given:
The transformations are
Reason:
Take the image of S in the uv- plane is
The uv- plane bounded by the vertices
From
Substitute
Therefore, xy- plane traces out the segment from
From
Substitute
Therefore, xy- plane traces out the segment from
From
Substitute
Therefore, xy- plane traces out the segment from
From
Substitute
Therefore, xy- plane traces out the segment from
Thus, the image of region in xy- plane is a square with vertices
Hence, it does not maps into a triangle and thereby the statement is not true.
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Chapter 13 Solutions
Calculus: Early Transcendentals, 2nd Edition
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