Exercise · ZW-FUBINIS-THEOREM-006 · ★★☆ Intermediate
Compute \(\iint_R 6xy^2\,dA\) over \(R=[0,1]\times[-1,1]\).
Choose one option.
Show answer
- C. \(2\)
Explanation
\(\int_0^1 6x\,dx=3\) and \(\int_{-1}^{1}y^2\,dy=\tfrac23\); the product is \(2\). It does not vanish: \(y^2\) is even.
Review the concept: Fubini's theorem
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-007 Let f be continuous on R=[a,b]×[c,d]. What does Fubini's theorem state?
- 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]?