Exercise · ZW-FUBINIS-THEOREM-007 · ★★☆ Intermediate
Let \(f\) be continuous on \(R=[a,b]\times[c,d]\). What does Fubini's theorem state?
Choose one option.
A That \(\iint_R f\,dA\) equals \(\int_a^b\!\int_c^d f\,dy\,dx\) and also \(\int_c^d\!\int_a^b f\,dx\,dy\). B That \(\iint_R f\,dA=\int_a^b f\,dx\cdot\int_c^d f\,dy\). C That \(\iint_R f\,dA=\int_a^b\!\int_c^d f\,dy\,dx\) only when integrating in \(y\) first. D That the two iterated integrals differ, but both approximate \(\iint_R f\,dA\). Show answer A. That \(\iint_R f\,dA\) equals \(\int_a^b\!\int_c^d f\,dy\,dx\) and also \(\int_c^d\!\int_a^b f\,dx\,dy\).← Back to the question
Explanation Both iterated integrals exist and equal the double integral . The product of integrals holds only if \(f=g(x)h(y)\).
If \(f\) is continuous on a rectangle \(R=[a,b]\times[c,d]\), its double integral with respect to the area \(A\) is computed as an iterated integral , integrating in either order.
\[\iint_R f\,dA=\int_a^b\!\int_c^d f(x,y)\,dy\,dx=\int_c^d\!\int_a^b f(x,y)\,dx\,dy\]
More practice problems: fubini's theorem ZW-FUBINIS-THEOREM-001 Compute iint_R xy dA over R=[0,2]×[0,3].ZW-FUBINIS-THEOREM-002 Compute iint_R (x+y) dA over R=[0,1]×[0,2].ZW-FUBINIS-THEOREM-003 Compute iint_R e^x+y dA over R=[0,1]×[0,1].ZW-FUBINIS-THEOREM-004 Compute iint_R x^2y dA over R=[1,2]×[0,1].ZW-FUBINIS-THEOREM-005 Compute iint_R sin xcos y dA over R=[0,tfracπ2]×[0,tfracπ2].ZW-FUBINIS-THEOREM-006 Compute iint_R 6xy^2 dA over R=[0,1]×[-1,1].ZW-FUBINIS-THEOREM-008 Compute iint_R xe^xy dA over R=[0,1]×[0,1]. It helps to integrate in y first.ZW-FUBINIS-THEOREM-009 If f(x,y)=g(x)h(y) with g,h continuous, what is iint_R f dA over R=[a,b]×[c,d]?