Definition: The fundamental theorem of calculus connects derivatives and integrals: differentiating an accumulation function returns the integrand, and a definite integral equals F(b) minus F(a).
What are the two parts?
- Part 1: if A(x) = ∫ from a to x of f(t) dt, then A'(x) = f(x).
- Part 2: ∫ from a to b of f(x) dx = F(b) – F(a) for any antiderivative F.
Together they say differentiation and integration undo each other.
How do you use Part 1 with a chain?
If the upper limit is a function of x, multiply by its derivative.
Worked example: find d/dx of ∫ from 1 to x2 of cos(t) dt. Part 1 gives cos(x2), and the chain rule adds the factor 2x. The answer is 2x cos(x2).
Part 2 example: ∫ from 0 to 2 of 3x2 dx = 23 – 0 = 8.
What mistake is common?
Forgetting the chain factor when the limit is not just x. Another is swapping the order of subtraction. Mathbot writes F(top) minus F(bottom) every time.
Example: d/dx of the integral from 1 to x^2 of cos t dt is 2x cos(x^2)