## What is volume integral in vector?

In mathematics (particularly multivariable calculus), a volume integral(∰) refers to an integral over a 3-dimensional domain; that is, it is a special case of multiple integrals. Volume integrals are especially important in physics for many applications, for example, to calculate flux densities.

## How do you find the volume of a toroid?

The volume of a torus is calculated by multiplying the area of the cross-section by the circumference of the ring. Volume = π × r2 × 2 × π × R.

**How do you integrate the volume of a torus?**

1 Answer. If the radius of its circular cross section is r , and the radius of the circle traced by the center of the cross sections is R , then the volume of the torus is V=2π2r2R .

**What is the difference between volume integral and surface integral?**

The Riemannian sum corresponding to a surface integral devides the surface into small squares (or other shape) and sums the value for those squares, while the volume integrals acts on a body and devides it into small cubes (or other 3-dimensional shape) and sums the values for those cubes.

### Which technique can be used to calculate the volume with only a single integral?

Definite integrals can be used to find the volumes of solids. Using the slicing method, we can find a volume by integrating the cross-sectional area. For solids of revolution, the volume slices are often disks and the cross-sections are circles.

### What is line integral surface integral and volume integral?

A line integral is an integral where the function to be integrated is evaluated along a curve and a surface integral is a generalization of multiple integrals to integration over surfaces. It can be thought of as the double integral analog of the line integral.

**How do you relate volume integral with surface integral?**

More precisely, the divergence theorem states that the surface integral of a vector field over a closed surface, which is called the flux through the surface, is equal to the volume integral of the divergence over the region inside the surface.