StellarThe Fundamental Theorem of Calculus, Part 2 says that if f is continuous on [a,b] and F is any antiderivative of f, then
This is the theorem that allows exact evaluation of definite integrals using antiderivatives instead of infinitely many rectangles. It completes the connection between differentiation and integration: Part 1 says differentiation reverses accumulation, and Part 2 says definite integrals can be computed through antiderivatives. This is one of the most important computational tools in the course.
To use this theorem, you first find an antiderivative of the integrand, then evaluate it at the upper and lower limits, and subtract. This process is sometimes called evaluating the antiderivative from a to b. It is important to keep the order correct: upper value minus lower value. This theorem turns integration from an approximation process into an exact method when an antiderivative can be found, and it becomes the basis for much of the rest of integral calculus.