How can interpolation be used to approximate integrals and derivatives?

In numerical analysis, interpolation serves as a foundational technique for approximating both integrals and derivatives of a function $f(x)$ when the function is known only at a discrete set of points $(x_i,y_i)$.1

The general process involves two primary steps:

• Construction: First, you construct an interpolating function $p(x)$—most commonly a polynomial—that passes through the known data points such that $p(x_i)=y_i$.21
• Approximation: Once the interpolating function is established, you perform the desired operation on $p(x)$ rather than the original, potentially more complex function $f(x)$.1

Specifically, to approximate an integral, you integrate the interpolating polynomial over the desired interval. To approximate a derivative, you differentiate the polynomial at the point of interest. Because polynomials are simple to evaluate and manipulate using basic arithmetic operations, they provide a tractable way to estimate these values when the exact form of $f(x)$ is unknown or difficult to compute directly.1