what is difference between numerical integration and interpolation?

Question

I am studying finite element method.While studying i am confuse with numerical integration and interpolation.Is this two methods are same or different?. If they are different then is there any relation between them?.

Answer 1

Interpolation is a problem to approximate the values of some function $f$ at some point $x$ if the values at $\{x_k\}$ are known.

Numerical integration is a technique to numerically approximate $\int_a^b f(x) dx$, given the values of $f(x)$ at some points.

They are not the same, but related.

Answer 2

Numerical integration provides an approximate value of an integral while interpolation creates a curve from a discrete signal. You can use interpolation to have a curve to work on whit numerical integration.

Answer 3

Interpolation finds a function which passes through a given function at specified points. Numerical integration finds an approximation for an integral which may not be analytically solvable.

There are many interpolation-based numerical integration techniques. Essentially, one chooses certain points of the interval, finds an interpolant at those points, and then integrates the interpolant. The interpolant is chosen from a class of functions which are easy to integrate. Piecewise polynomials are the most common example.

The composite trapezoidal rule is a concrete example. This rule breaks the interval up into $N$ pieces of equal size, finds a linear function which agrees with $f$ at the endpoints of each piece, and integrates the resulting piecewise linear function. Rules using higher degree polynomials, different choices of the interpolation nodes, and even functions other than polynomials can be developed.

On the other hand, there are numerical integration techniques which do not use interpolation at all. Gaussian quadrature is a major example.