Multivariate Calculus // Integration

MULTIPLE
INTEGRALS

To calculate the area under a 2D line, you use a standard integral. To calculate the physical volume under a 3D surface, you must integrate twice. It is the mathematical act of sweeping an area through space to construct solid geometry.

From Area to Volume

In single-variable calculus, we approximated area by packing thin 2D rectangles under a curve (a Riemann sum) and taking the limit as the rectangles became infinitely thin.

In multivariate calculus, our domain is no longer a simple line segment on the X-axis; it is a flat 2D grid ( $R$ ) on the XY-plane. Instead of drawing 2D rectangles up to the curve, we must pull 3D rectangular columns up to the surface $f(x,y)$ .

V \approx \sum_{i=1}^{n} \sum_{j=1}^{m} f(x_i, y_j) \Delta A

Fubini's Theorem

As our columns become infinitely thin, our summation transforms into a Double Integral. But how do we actually calculate it? Fubini's Theorem provides the elegant solution: we slice the volume.

If you calculate the 2D cross-sectional area of a single "slice" of the volume by integrating along the X-axis, you can then integrate that resulting area equation along the Y-axis to sweep out the total volume.

V = \iint_{R} f(x,y) dA = \int_{c}^{d} \left( \int_{a}^{b} f(x,y) dx \right) dy

3D Riemann Sums

Discrete Volume Approximation

Grid Resolution (N = 6)Higher N approaches true volume

Notice how small values of N leave jagged, inaccurate gaps. As you drag the slider to the right, the discrete rectangular prisms shrink, seamlessly packing together to approach the true, smooth continuous surface of the mathematical function.

Advanced Integration Domains

What happens when your domain isn't a perfect square grid?

Polar Coordinates & Jacobians

Changing variables to integrate over circular, cylindrical, or spherical domains.

Triple Integrals

Integrating a 3D volume to find hyperspace mass, density, and centers of gravity.