Training Multivariable Calculus Multiple Integrals
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Multiple Integrals

26 min Multivariable Calculus

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

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

Outer: $\int_0^1 (2x+2)\,dx = [x^2+2x]_0^1 = 3$.

Example 2

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

Inner: $\int_0^x x^2y\,dy = x^2 \cdot x^2/2 = x^4/2$.

Outer: $\int_0^1 x^4/2\,dx = \frac{1}{10}$.

Example 3

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

$$A = \int_0^2 \int_0^x 1\,dy\,dx = \int_0^2 x\,dx = 2$$

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

Partial Derivatives Gradient & Directional Derivatives