Exercise · ZW-DOUBLE-INTEGRAL-OVER-GENERAL-REGION-008 · ★★☆ Intermediate
Find the area of \(D=\{1\le x\le e,\ 0\le y\le\ln x\}\).
Choose one option.
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- C. \(1\)
Explanation
The area is \(\int_1^e\ln x\,dx=\left[x\ln x-x\right]_1^e=0-(-1)=1\).
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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- 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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