Multivariable calculus

Double integrals

Volume under a surface, and the order the bounds have to go in.

• Double Integral of xy Over [0,2] by [0,3]The double integral of xy over the rectangle 0 to 2 in x and 0 to 3 in y is 9. The iterated setup, the separable shortcut, and the limits mistake.
• Double Integral of x^2 + y^2 Over [0,1] by [0,2]The double integral of x^2 + y^2 over the rectangle 0 to 1 in x and 0 to 2 in y is 10/3. Why a sum splits by linearity but never factors.
• Double Integral of (x + y)^2 Over the Unit SquareThe double integral of (x + y)^2 over the unit square is 7/6. Integrate with a shift substitution or expand first, and see why 1 is the wrong answer.
• Double Integral of sin x cos y Over a pi/2 SquareThe double integral of sin x cos y over the square 0 to pi/2 in both variables is 1, because the integrand factors into one function of x and one of y.
• Double Integral of sin(x + y) Over a pi/2 SquareThe double integral of sin(x + y) over the square 0 to pi/2 in both variables is 2. The angle sum blocks the factoring shortcut, and the identity fixes it.
• Double Integral of e^(x + y) Over the Unit SquareThe double integral of e^(x + y) over the unit square is (e - 1)^2, about 2.95249. The exponent law makes this one factor, unlike sin(x + y).
• Double Integral of x e^(xy) Over [0,1] by [0,2]The double integral of x e^(xy) over x in 0 to 1 and y in 0 to 2 is (e^2 - 3)/2, about 2.19453. Integrate in y first and no integration by parts is needed.
• Double Integral of y sin(xy) Over [1,2] by [0, pi/2]The double integral of y sin(xy) with x from 1 to 2 and y from 0 to pi/2 is 1. Integrating in x first turns the leading y into the substitution.
• Double Integral of y/(1 + x^2) Over [0,1] by [0,2]The double integral of y/(1 + x^2) over x in 0 to 1 and y in 0 to 2 is pi/2. It factors into an arctangent times a simple power integral.
• Double Integral of 1/(x + y) Over the Square [1,2]The double integral of 1/(x + y) over the square 1 to 2 in both variables is 10 ln 2 minus 6 ln 3, about 0.33980. Two logarithms, then a log integral.
• Double Integral of xy e^(-x^2) Over [0,1] by [0,2]The double integral of xy e^(-x^2) over x in 0 to 1 and y in 0 to 2 is 1 minus 1/e, about 0.63212. The loose x is what makes the substitution work.
• Double Integral of x^3 y^2 Over [-1,2] by [0,2]The double integral of x^3 y^2 over x from -1 to 2 and y from 0 to 2 equals 10. A negative lower limit makes part of the volume count negatively.
• Double Integral of x e^(xy) Over a RectangleThe double integral of x e^(xy) over the rectangle 0 to 1 by 0 to 2 is (e^2 - 3)/2. Integrate in y first, and see why the other order needs parts.
• Double Integral of x^2 sin y Over a RectangleThe double integral of x^2 sin(y) over 0 <= x <= 1, 0 <= y <= pi equals 2/3. A product integrand on a rectangle splits into two single integrals.
• Double Integral of xy Over a Triangle: 1/8The double integral of xy over the triangle bounded by y = 0, y = x and x = 1 equals 1/8. The type 1 setup, the square trap, and a check by reversing.
• Double Integral of x + y Over the Unit TriangleThe double integral of x + y over the triangle x + y <= 1 in the first quadrant equals 1/3. Full setup, the binomial trap, and a symmetry check.
• Double Integral of x^2 y Under y = x^2The double integral of x^2 y over the region 0 <= y <= x^2 with 0 <= x <= 2 equals 64/7. Type 1 setup, the squaring trap, and the reversed order check.
• Double Integral of y Between y = x^2 and y = xThe double integral of y over the region between y = x^2 and y = x equals 1/15. Which curve is on top, the full setup, and the centroid check.
• Double Integral of x Under y = sin x from 0 to 1The double integral of x over the region under y = sin x for 0 <= x <= 1 equals sin 1 - cos 1, about 0.301169. Setup, parts, and the sign trap.
• Double Integral of x/(1+y^2) Over a TriangleThe double integral of x/(1 + y^2) over the triangle with vertices (0,0), (2,0), (2,1) equals pi - 2. The arctangent step, parts, and the easier order.
• Double Integral of y^2 Between x = y^2 and x = sqrt yThe double integral of y^2 over the region between x = y^2 and x = sqrt(y) equals 3/35. The type 2 setup, which curve is on the right, and a check.
• Double Integral of e^(y^2) Over a TriangleThe double integral of e^(y^2) over the triangle 0 <= x <= y <= 1 equals (e - 1)/2. Only one order of integration is elementary, and here is why.
• Double Integral of y Between x = y and x = 2 - y^2The double integral of y over the region between x = y and x = 2 - y^2 equals -9/4. Why type 2 avoids a split, and why the answer is negative.
• Double Integral of 1/x Between x = y and x = y^2The double integral of 1/x over the region between x = y and x = y^2 for 1 <= y <= 2 equals 2 ln 2 - 1. The log collapse and the identity trap.
• Double Integral of sin x cos y Over a SquareThe double integral of sin x cos y over the square from 0 to pi/2 in both variables equals 1. Fubini step by step, plus the separable shortcut.
• Double Integral of exp(x + y) Over the Unit SquareThe double integral of e^(x+y) over the unit square equals (e-1)^2, about 2.9525. See the Fubini steps and the exponent-splitting error to avoid.
• Double Integral of x exp(xy) Over the Unit SquareThe double integral of x e^(xy) over the unit square equals e - 2, about 0.7183. Integrating in y first avoids integration by parts entirely.
• Double Integral of y sin(xy) Over a RectangleThe double integral of y sin(xy) with x from 0 to pi/2 and y from 0 to 1 equals 1 - 2/pi, about 0.3634. Integrating in x first avoids parts.
• Double Integral of exp(y^2) Over a TriangleThe double integral of e^(y^2) over the triangle 0 to y in x, 0 to 1 in y, equals (e-1)/2. The classic case where only one order of integration works.
• Double Integral of y sin x Under a Cosine ArchThe double integral of y sin x over the region under y = cos x from x = 0 to pi/2 equals 1/6. Type 1 bounds, with the variable ceiling worked through.
• Double Integral of exp(-x-y) Over a TriangleThe double integral of e^(-x-y) over the triangle x + y at most 1 in the first quadrant equals 1 - 2/e, about 0.2642. Full type 1 setup and working.
• Double Integral of sin(x + y) Over a SquareThe double integral of sin(x + y) over the square from 0 to pi/2 in both variables equals 2. Worked twice over, with the sin(x+y) split error explained.
• Double Integral of sin^2 x cos^2 y Over a SquareThe double integral of sin^2 x cos^2 y over the square from 0 to pi in both variables equals pi^2/4. Half-angle identities and the power-rule trap.
• Double Integral of cos(x^2) Over a TriangleThe double integral of cos(x^2) over the triangle 0 to x in y, 0 to sqrt(pi/2) in x, equals 1/2. Only one order of integration is elementary.