Exercise · ZW-DOUBLE-INTEGRAL-OVER-GENERAL-REGION-010 · ★★★ Advanced
Compute \(\int_0^1\!\int_x^1 e^{y^2}\,dy\,dx\). Reverse the order of integration.
Choose one option.
Show answer
- C. \(\tfrac{e-1}{2}\)
Explanation
The region is \(0\le x\le y\le1\), so the integral is \(\int_0^1\!\int_0^y e^{y^2}\,dx\,dy=\int_0^1ye^{y^2}\,dy=\tfrac{e-1}{2}\). In the original order, \(e^{y^2}\) has no elementary antiderivative.
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.
- 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\.