Why does the fundamental theorem of calculus work?
The fundamental theorem of calculus says differentiation and integration undo each other. The area under f from a fixed point up to x grows at the rate f(x), so the area from a to b equals F(b) − F(a) for any antiderivative F of f.
The two parts
- Part 1: if f is continuous and A(x) = ∫ from a to x of f(t) dt, then A′(x) = f(x). The accumulated area is an antiderivative of f.
- Part 2: if F′(x) = f(x) on [a, b], then ∫ from a to b of f(x) dx = F(b) − F(a).
Part 1 is the reason; Part 2 is the tool you use most. Together they mean you can find an area, which looks like an infinite sum of thin rectangles, by finding a function whose derivative is f and subtracting two values.
Why the area grows at the rate f(x)
Shade the area under f from a up to x and call it A(x). Now push x forward by a small amount Δx. The new area is the old area plus a thin strip about f(x) tall and Δx wide, so A(x + Δx) − A(x) ≈ f(x) · Δx. Divide by Δx and you get a rate of change close to f(x). As Δx shrinks to zero the approximation becomes exact, and that rate is the derivative of A. The taller the curve where you are, the faster the area piles up.
A worked integral and the usual traps
Find ∫ from 1 to 3 of x² dx. An antiderivative is F(x) = x³/3, so the answer is F(3) − F(1) = 9 − 1/3 = 26/3 ≈ 8.67. As a check, two midpoint rectangles of width 1 at x = 1.5 and x = 2.5 give 2.25 + 6.25 = 8.5, close to the exact value.
Part 1 also shows up with a variable limit. For d/dx of ∫ from 0 to x² of cos(t) dt, plug x² into the integrand and multiply by the inner derivative: cos(x²) · 2x. That is the chain rule at work.
- Forgetting the chain rule factor when the upper limit is a function of x.
- Computing only F(b) and dropping F(a).
- Treating a definite integral as total area. It is net signed area: parts below the x-axis count as negative.
- Using it on a function that is not continuous on the interval. ∫ from −1 to 1 of 1/x² dx cannot be done this way, because 1/x² blows up at 0.
Seeing the strip
Type "Why does the fundamental theorem of calculus work?" into Isotoon and you get a short animated video built around that strip: the area filling in under the curve, the edge moving forward, and the strip's height matching f(x). On Pro you can also ask for a 2D interactive and drag x yourself. More on the course is on AP Calculus and calculus.
Questions
- Why don't you need +C in a definite integral?
- Any constant you add to F appears in both F(b) and F(a), so it cancels when you subtract. Every antiderivative gives the same answer.
- What is the difference between Part 1 and Part 2?
- Part 1 says the accumulation function A(x), the integral from a to x of f, has derivative f(x). Part 2 uses that fact to evaluate a definite integral as F(b) minus F(a) for any antiderivative F.