Antiderivatives Unlock the Integral
The Fundamental Theorem says the two great ideas of calculus, slopes and areas, are inverse operations.
Defining the Integral
The definite integral of f from a to b is the limit of Riemann sums as the slices get infinitely thin. It measures signed area: regions above the axis count positive, regions below count negative. A function can be negative everywhere and still have a perfectly good, negative integral.
\int_a^b f(x)\,dx = \lim_{n\to\infty}\sum_{i=1}^{n} f(x_i^*)\,\Delta xFTC Part 1
Define an area-so-far function by integrating f from a fixed point a up to a variable endpoint x. Differentiate that accumulation and you get back f itself. This is the theorem's first half: integration, run forward, is undone by differentiation.
\frac{d}{dx}\int_a^x f(t)\,dt = f(x)FTC Part 2
The second half says you can evaluate a definite integral with any antiderivative F: the answer is F(b) minus F(a), no limits required. For f(x) = x^2 on [0, 2], an antiderivative is x^3/3, so the integral is 8/3 minus 0. What took an infinite sum by hand is now two evaluations.
\int_a^b f(x)\,dx = F(b) - F(a)Flashcard
Evaluation form of the Fundamental Theorem
Check your understanding
A car's velocity is v(t) = 2t for 0 to 3 seconds. What does the integral of v from 0 to 3 give, and what is its value?