Exercise · ZW-DOUBLE-INTEGRAL-OVER-GENERAL-REGION-007 · ★★☆ Intermediate
Compute \(\iint_D xy\,dA\), where \(D=\{0\le y\le1,\ y\le x\le2\}\).
Choose one option.
Show answer
- D. \(\tfrac78\)
Explanation
With \(x\) inside: \(\int_y^2xy\,dx=y\left(2-\tfrac{y^2}{2}\right)\), and \(\int_0^1\left(2y-\tfrac{y^3}{2}\right)dy=1-\tfrac18=\tfrac78\).
Review the concept: Double integral over a general region
The double integral of \(f\) with respect to the area \(A\) over a region \(D=\{a\le x\le b,\ g_1(x)\le y\le g_2(x)\}\) is an iterated integral whose inner limits depend on the outer variable.
\[\iint_D f\,dA=\int_a^b\!\int_{g_1(x)}^{g_2(x)} f(x,y)\,dy\,dx\]
More practice problems: double integral over a general region
- ZW-DOUBLE-INTEGRAL-OVER-GENERAL-REGION-001 Compute iint_D (x+y) dA, where D=\0≤ x≤1,\ 0≤ y≤ x\.
- ZW-DOUBLE-INTEGRAL-OVER-GENERAL-REGION-002 Compute iint_D 2xy dA, where D=\0≤ x≤1,\ x^2≤ y≤ x\.
- ZW-DOUBLE-INTEGRAL-OVER-GENERAL-REGION-003 Find the area of the region between y=x^2 and y=2x, that is, iint_D dA.
- ZW-DOUBLE-INTEGRAL-OVER-GENERAL-REGION-004 Compute iint_D x dA, where D=\0≤ x≤2,\ 0≤ y≤ x^2\.
- ZW-DOUBLE-INTEGRAL-OVER-GENERAL-REGION-005 The region D=\0≤ y≤ x≤1\ can also be described by integrating in x first. Which iterated integral e…
- ZW-DOUBLE-INTEGRAL-OVER-GENERAL-REGION-006 Compute ∫_0^1\!∫_y^1 sin (x^2) dx dy by reversing the order of integration.
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- ZW-DOUBLE-INTEGRAL-OVER-GENERAL-REGION-009 Let D be the region between the graphs y=g_1(x) and y=g_2(x), with g_1≤ g_2 and a≤ x≤ b. Which is t…
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