Training Multivariable Calculus Multiple Integrals
2 / 5

Multiple Integrals

26 min Multivariable Calculus
Multiple integrals generalize single-variable integration to compute volumes, masses, and accumulated quantities over regions in the plane or in space. A double integral ∬_R f(x, y) dA sums the function f over a two-dimensional region R, typically evaluated as an iterated integral by integrating first with respect to one variable and then the other. Choosing the correct order of integration—and sometimes switching to polar, cylindrical, or spherical coordinates—can dramatically simplify the computation. Applications range from finding the area of irregular regions and the center of mass of a lamina to computing probabilities over joint distributions.

Multiple Integrals

Double Integral

$$\iint_R f(x,y)\,dA = \int_a^b \int_{g_1(x)}^{g_2(x)} f(x,y)\,dy\,dx$$

Gives the volume under the surface $z = f(x,y)$ over region $R$.

Iterated Integrals

Evaluate from inside out — integrate the inner variable first, treating the outer variable as constant.

Triple Integral

$$\iiint_E f(x,y,z)\,dV = \int_a^b\int_{g_1}^{g_2}\int_{h_1}^{h_2} f\,dz\,dy\,dx$$

Example 1

Evaluate $\int_0^1 \int_0^2 (x+y)\,dy\,dx$.

  1. Inner: $\int_0^2 (x+y)\,dy = [xy + y^2/2]_0^2 = 2x + 2$.
  2. Outer:
  3. $\int_0^1 (2x+2)\,dx = [x^2+2x]_0^1 = 3$.
Example 2

Evaluate $\int_0^1 \int_0^x x^2y\,dy\,dx$.

  1. Inner: $\int_0^x x^2y\,dy = x^2 \cdot x^2/2 = x^4/2$.
  2. Outer:
  3. $\int_0^1 x^4/2\,dx = \frac{1}{10}$.
Example 3

Find the area of $R$: $0 \leq x \leq 2$, $0 \leq y \leq x$.

  1. Set up the double or triple integral.
  2. $$A = \int_0^2 \int_0^x 1\,dy\,dx = \int_0^2 x\,dx = 2$$

Practice Problems

1. Evaluate $\int_0^2\int_0^3 xy\,dy\,dx$.
2. Evaluate $\int_0^1\int_0^1 (x^2+y^2)\,dy\,dx$.
3. Evaluate $\int_0^{\pi}\int_0^1 r\,dr\,d\theta$ (polar).
4. Set up the integral for the area under $y = x^2$ from $x=0$ to $x=1$.
5. Evaluate $\int_0^1\int_0^{2x} 1\,dy\,dx$.
6. Switch order: $\int_0^1\int_0^x f\,dy\,dx$.
7. Evaluate $\int_0^2\int_0^1\int_0^3 1\,dz\,dy\,dx$.
8. Evaluate $\int_0^1\int_x^1 e^{y^2}\,dy\,dx$ by switching order.
9. Find the volume under $z = 4-x^2-y^2$ over $R: 0 \leq x \leq 1, 0 \leq y \leq 1$.
10. Evaluate $\int_0^1\int_0^y 2xy\,dx\,dy$.
11. What does $\iint_R 1\,dA$ represent?
12. Mass of plate: density $\rho = xy$, $R: [0,2]\times[0,3]$.
Show Answer Key

1. $\int_0^2 \frac{9x}{2}\,dx = 9$

2. $\int_0^1(x^2+1/3)\,dx = 1/3+1/3 = 2/3$

3. $\int_0^{\pi} 1/2\,d\theta = \pi/2$

4. $\int_0^1\int_0^{x^2} 1\,dy\,dx$

5. $\int_0^1 2x\,dx = 1$

6. $\int_0^1\int_y^1 f\,dx\,dy$

7. $2 \cdot 1 \cdot 3 = 6$

8. $\int_0^1\int_0^y e^{y^2}\,dx\,dy = \int_0^1 ye^{y^2}\,dy = \frac{e-1}{2}$

9. $\int_0^1\int_0^1 (4-x^2-y^2)\,dy\,dx = 4-1/3-1/3 = 10/3$

10. $\int_0^1 y^3\,dy = 1/4$

11. The area of region $R$

12. $\int_0^2\int_0^3 xy\,dy\,dx = \int_0^2 9x/2\,dx = 9$

∬ Double Integral Estimator
∫₀ᵃ xⁿ dx
∫₀ᵇ yᵐ dy
∬ xⁿyᵐ dA
Partial Derivatives Gradient & Directional Derivatives