Let R = [a, b] × [c, d] be a rectangle. Also suppose that f (x) is continuous on [a, b] and that g(y) is continuous on [c, d]. Show that qu //, f(2)g(y) dA = f (x) dx g(y) dy R a Use your answer to part (a) to evaluate xy + x + y + 1 dA, where R R= [0, 1] x [0, 1]. (Hint: Factor the integrand.)
Let R = [a, b] × [c, d] be a rectangle. Also suppose that f (x) is continuous on [a, b] and that g(y) is continuous on [c, d]. Show that qu //, f(2)g(y) dA = f (x) dx g(y) dy R a Use your answer to part (a) to evaluate xy + x + y + 1 dA, where R R= [0, 1] x [0, 1]. (Hint: Factor the integrand.)
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![Let \( R = [a, b] \times [c, d] \) be a rectangle. Also suppose that \( f(x) \) is continuous on \([a, b]\) and that \( g(y) \) is continuous on \([c, d]\). Show that
\[
\iint_R f(x)g(y) \, dA = \left( \int_a^b f(x) \, dx \right) \left( \int_c^d g(y) \, dy \right)
\]
Use your answer to part (a) to evaluate
\[
\iint_R xy + x + y + 1 \, dA,
\]
where \( R = [0, 1] \times [0, 1] \). (Hint: Factor the integrand.)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F66e8a441-1535-4b77-8170-ec0b022e4104%2Fad81b34e-2b1d-49d9-a376-af435684bee6%2Fsy8iq7_processed.png&w=3840&q=75)
Transcribed Image Text:Let \( R = [a, b] \times [c, d] \) be a rectangle. Also suppose that \( f(x) \) is continuous on \([a, b]\) and that \( g(y) \) is continuous on \([c, d]\). Show that
\[
\iint_R f(x)g(y) \, dA = \left( \int_a^b f(x) \, dx \right) \left( \int_c^d g(y) \, dy \right)
\]
Use your answer to part (a) to evaluate
\[
\iint_R xy + x + y + 1 \, dA,
\]
where \( R = [0, 1] \times [0, 1] \). (Hint: Factor the integrand.)
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