Exercise · ZW-DOUBLE-INTEGRAL-OVER-GENERAL-REGION-009 · ★★★ Advanced
Let \(D\) be the region between the graphs \(y=g_1(x)\) and \(y=g_2(x)\), with \(g_1\le g_2\) and \(a\le x\le b\). Which is the correct iterated integral of \(f\) over \(D\)?
Choose one option.
A \(\int_{g_1(x)}^{g_2(x)}\!\int_a^b f\,dy\,dx\) B \(\int_a^b\!\int_{g_1(y)}^{g_2(y)} f\,dy\,dx\) C \(\int_{g_1(a)}^{g_2(b)}\!\int_a^b f\,dx\,dy\) D \(\int_a^b\!\int_{g_1(x)}^{g_2(x)} f\,dy\,dx\) Show answer D. \(\int_a^b\!\int_{g_1(x)}^{g_2(x)} f\,dy\,dx\)← Back to the question
Explanation Outer limits are constants and inner limits may depend on the outer variable: here \(y\) lies between \(g_1(x)\) and \(g_2(x)\).
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.ZW-DOUBLE-INTEGRAL-OVER-GENERAL-REGION-007 Compute iint_D xy dA, where D=\0≤ y≤1,\ y≤ x≤2\.ZW-DOUBLE-INTEGRAL-OVER-GENERAL-REGION-008 Find the area of D=\1≤ x≤ e,\ 0≤ y≤ln x\.