Exercise · ZW-DOUBLE-INTEGRAL-OVER-GENERAL-REGION-002 · ★☆☆ Basic
Compute \(\iint_D 2xy\,dA\), where \(D=\{0\le x\le1,\ x^2\le y\le x\}\).
Choose one option.
Show answer
- B. \(\tfrac1{12}\)
Explanation
\(\int_{x^2}^{x}2xy\,dy=x(x^2-x^4)=x^3-x^5\), and \(\int_0^1(x^3-x^5)\,dx=\tfrac14-\tfrac16=\tfrac1{12}\).
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
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