In calculus, finding the area under a curve is a fundamental concept that allows us to calculate the 'signed' area enclosed by a function and the x-axis. When it comes to polynomial functions, the process is similar to what we learned for linear functions, but with a slight twist.

Let's consider a polynomial function, f(x) = ax^n + bx^(n-1) + cx^(n-2) + ... + d, where a, b, c, d, and n are constants. To find the area under this polynomial curve, we can use definite integration. The definite integral of a function f(x) over an interval [a, b] represents the signed area between the curve and the x-axis within that interval.

To illustrate this, let's take an example. Suppose we have a quadratic function f(x) = x^2 - 2x + 1. To find the area under this curve between x = 0 and x = 2, we can integrate f(x) with respect to x using definite integration: ∫[0,2] (x^2 - 2x + 1) dx. Evaluating this integral gives us the desired area under the polynomial curve.

It's important to note that when the polynomial function intersects or lies below the x-axis within the interval of integration, the 'signed' area will be negative. This indicates that the enclosed area is below the x-axis. To calculate the actual area, we take the absolute value of the signed area.

Remember, when finding the area under polynomial functions, integrate the function with respect to x over the desired interval. Consider the sign of the result to interpret if the area is positive or negative.