How is a definite integral evaluated?

A definite integral is evaluated by finding the antiderivative of the function being integrated and then applying the limits of integration. This process is grounded in the Fundamental Theorem of Calculus, which establishes a crucial connection between differentiation and integration.

When you find the antiderivative of a function, you're essentially determining a new function, which gives you the area under the curve of the original function up to a certain point. Once you have this antiderivative, you substitute the upper and lower limits of the integral into it. The difference between these two values gives the exact area under the curve between the specified bounds.

For instance, if we have an integral from (a) to (b) of a function (f(x)), you would first calculate (F(b) - F(a)) where (F) is the antiderivative of (f). This process yields the total signed area between the curve of (f), the x-axis, and the vertical lines at (x = a) and (x = b).

In contrast, approximating areas under curves typically refers to numerical methods like Riemann sums or trapezoidal rules, which are not exact evaluations of definite integrals. Summ