Double Integral: Step-by-Step Guide
A double integral computes the signed volume under a surface z = f(x, y) over a 2D region R. It is evaluated as an iterated integral: first integrate with respect to one variable (treating the other as constant), then integrate the result with respect to the second variable.
Math Racer skill guide · Calculus 3 · College
How to solve it, step by step
- Sketch the region of integration. Draw R in the xy-plane. Identify whether it is more naturally described as Type I (vertical strips, dy dx) or Type II (horizontal strips, dx dy).\[R = \{(x,y) : a \le x \le b,\; g_1(x) \le y \le g_2(x)\}\]
- Set up the iterated integral. Write the double integral as a nested integral with the inner limits possibly depending on the outer variable.\[\iint_R f(x,y)\,dA = \int_a^b \int_{g_1(x)}^{g_2(x)} f(x,y)\,dy\,dx\]
- Evaluate the inner integral. Treat the outer variable as constant. Find an antiderivative with respect to the inner variable and evaluate at the inner limits.\[\int_{g_1(x)}^{g_2(x)} f(x,y)\,dy = G(x)\]
- Evaluate the outer integral. Integrate the result G(x) with respect to the outer variable over its constant limits.\[\int_a^b G(x)\,dx\]
- Switch order if useful. If the inner integral is too hard, redescribe R as Type II and swap dx dy. The numeric value is the same.\[\int_a^b\!\!\int_{g_1}^{g_2} = \int_c^d\!\!\int_{h_1}^{h_2}\]
Worked example
Evaluate the double integral over R of (x + y) dA, where R is the rectangle 0 <= x <= 2, 0 <= y <= 3.
Step 1: R is a rectangle, so limits are constants. Set up as integral from x=0 to 2 of integral from y=0 to 3 of (x + y) dy dx.
Step 2: Evaluate the inner integral with respect to y, treating x as constant: integral from 0 to 3 of (x + y) dy = [xy + y^2/2] from 0 to 3 = 3x + 9/2.
Step 3: Evaluate the outer integral: integral from 0 to 2 of (3x + 9/2) dx = [3x^2/2 + 9x/2] from 0 to 2 = (6 + 9) - 0 = 15.
Step 4: The double integral equals 15.
Verification: Switch order. Inner: integral from 0 to 2 of (x + y) dx = [x^2/2 + xy] from 0 to 2 = 2 + 2y. Outer: integral from 0 to 3 of (2 + 2y) dy = [2y + y^2] from 0 to 3 = 6 + 9 = 15. Matches.
Tips and common mistakes
- Always sketch R first; the limits depend on its shape.
- Inner limits may depend on the outer variable; outer limits must be constants.
- Integrate with respect to the inner variable first, treating the outer variable as constant.
- If the inner integral has no elementary antiderivative, swap the order of integration.
- For rectangular regions with separable integrand f(x,y) = g(x)h(y), the double integral factors into a product of single integrals.
- Convert to polar coordinates when R has circular symmetry: dA = r dr d(theta).
- Always check by swapping the order; both orders should give the same answer.
- Don't forget the dA element when writing the integral.