Approximate Calculation of a Definite Integral
Big Idea: Many definite integrals cannot be evaluated exactly using antiderivatives. In such cases, we use numerical methods to approximate the value of $\int_a^b f(x)\,dx$ by breaking the area into simple geometric shapes.
Why Approximate?
Consider the integral $\int_0^1 e^{-x^2}\,dx$. The function $e^{-x^2}$ has no elementary antiderivative! Yet this integral appears in probability, physics, and engineering. Numerical approximation methods allow us to compute such integrals to any desired accuracy.