📖 15. Multiple Integrals Calculus

📖 From Calculus • 636 questions available

About 15. Multiple Integrals Calculus

Definition:
Multiple integrals extend the concept of a single integral to functions of several variables, allowing us to integrate over regions in two or more dimensions, such as areas, volumes, or higher-dimensional spaces. For a function $f(x,y)$, the double integral over a region $R$ is written as $\iint_R f(x,y) \, dA$, and for three variables, the triple integral is $\iiint_V f(x,y,z) \, dV$.

Example:
Compute the volume under the surface $f(x,y) = 4 - x^2 - y^2$ over the circular region $R: x^2 + y^2 \leq 4$. In polar coordinates, this becomes $\int_0^{2\pi} \int_0^2 (4 - r^2) \, r \, dr \, d\theta = \int_0^{2\pi} \left[ 2r^2 - \frac{r^4}{4} \right]_0^2 d\theta = \int_0^{2\pi} (8 - 4) \, d\theta = 8\pi$.

Reason:
Multiple integrals are essential for computing physical quantities like area, volume, mass, center of mass, and total charge over non-rectangular domains, where single-variable integration is insufficient. They also enable change of variables (e.g., polar, cylindrical, spherical) to simplify complex regions, making them foundational in physics, engineering, and probability.

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🔄 Last updated: 2026-08-19