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Multiple Integrals

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Core Definition

Integrals of multivariable functions over 2D regions (double integrals) or 3D regions (triple integrals). A double integral is defined as a limit of double Riemann sums:

∬Df(x,y) dA=lim⁑ΔAiβ†’0βˆ‘if(xiβˆ—,yiβˆ—)Ξ”Ai\iint_{D} f(x, y) \, dA = \lim_{\Delta A_i \to 0} \sum_{i} f(x_i^*, y_i^*) \Delta A_i

Concept Explanation

Multiple integrals accumulate quantities over geometric domains. A double integral of a surface z=f(x,y)z = f(x, y) over a region RR evaluates the volume under the surface. Integrating over polar, cylindrical, or spherical coordinates simplifies calculations for rotationally symmetric domains.

Visual / Geometric Intuition

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Engineering Applications

Engineers use multiple integrals to find spatial averages, compute the center of gravity and moments of inertia for irregular structural components, and calculate total mass of variable density materials.

Example Problem

Evaluate the double integral:

∬Rxy dAwhereΒ R=[0,2]Γ—[1,3]\iint_{R} x y \, dA \quad \text{where } R = [0, 2] \times [1, 3]

Solution

Using Fubini's Theorem, convert the double integral into iterated integrals:

∬Rxy dA=∫02∫13xy dy dx\iint_{R} x y \, dA = \int_{0}^{2} \int_{1}^{3} x y \, dy \, dx

Step 1: Integrate the inner integral with respect to yy

∫13xy dy=x[12y2]13=x(92βˆ’12)=4x\int_{1}^{3} x y \, dy = x \Big[ \frac{1}{2} y^2 \Big]_1^3 = x \left( \frac{9}{2} - \frac{1}{2} \right) = 4x

Step 2: Integrate the outer integral with respect to xx

∫024x dx=[2x2]02=2(4)βˆ’2(0)=8\int_{0}^{2} 4x \, dx = \Big[ 2x^2 \Big]_0^2 = 2(4) - 2(0) = 8

Connections & References