Exercise · ZW-DOUBLE-INTEGRAL-OVER-GENERAL-REGION-004 · ★☆☆ Basic
Compute \(\iint_D x\,dA\), where \(D=\{0\le x\le2,\ 0\le y\le x^2\}\).
Choose one option.
Show answer
- B. \(4\)
Explanation
The inner integral is \(\int_0^{x^2}x\,dy=x^3\), and \(\int_0^2x^3\,dx=4\). Forgetting the factor \(x\) would give \(\tfrac83\).
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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