Exercise · ZW-DOUBLE-INTEGRAL-OVER-GENERAL-REGION-006 · ★★☆ Intermediate
Compute \(\int_0^1\!\int_y^1 \sin(x^2)\,dx\,dy\) by reversing the order of integration.
Choose one option.
Show answer
- A. \(\tfrac{1-\cos 1}{2}\)
Explanation
The region is \(0\le y\le x\le1\), so the integral is \(\int_0^1\!\int_0^x\sin(x^2)\,dy\,dx=\int_0^1x\sin(x^2)\,dx=\tfrac{1-\cos1}{2}\).
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-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\.
- 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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