Integrals & the fundamental theorem of calculus
The idea
A definite integral is the limit of Riemann sums: chop an interval into slivers, multiply each sliver's width by a function value, add, and refine. It computes accumulated change — area under a curve, distance from velocity, total charge from current. You already know area formulas for rectangles and triangles; integration is the upgrade that handles curved boundaries by treating area as a sum of infinitely many infinitely thin rectangles.
The fundamental theorem of calculus is the reason this is computable: if F is any antiderivative of f (meaning F'(x) = f(x)), then the integral of f from a to b equals F(b) − F(a). Differentiation and integration are inverse processes, so accumulating a rate recovers total change. The model to internalize: integrand = rate, integral = net amount.
Two misconceptions to retire. First, the definite integral is signed — regions below the x-axis count negatively, so an integral of zero can hide large areas that cancel. Second, the '+ C' belongs to indefinite integrals only; in a definite integral the constant subtracts away in F(b) − F(a), which is why no one writes it there.
Worked example
Evaluate the integral of f(x) = 6x² − 4x + 3 from x = 1 to x = 3 using the fundamental theorem of calculus.
- Find an antiderivative term by term with the reversed power rule: F(x) = 2x³ − 2x² + 3x, since 6x² antidifferentiates to 2x³, −4x to −2x², and 3 to 3x.
- Verify by differentiating back: F'(x) = 6x² − 4x + 3, which matches the integrand exactly — always worth the five seconds.
- Apply the fundamental theorem: the integral equals F(3) − F(1).
- Compute each piece: F(3) = 2(27) − 2(9) + 9 = 54 − 18 + 9 = 45, and F(1) = 2 − 2 + 3 = 3.
- Subtract: 45 − 3 = 42. As a plausibility check, the integrand runs from 5 at x = 1 up to 45 at x = 3, so an average height around 21 over a width of 2 is entirely consistent with 42.
Answer. The integral of 6x² − 4x + 3 from 1 to 3 equals 42 square units of accumulated area.
Check your understanding
- Why does any antiderivative work in F(b) − F(a), when antiderivatives differ by a constant?
- How does the Riemann-sum picture explain why regions below the x-axis contribute negatively?
- What real quantity does the integral compute when the integrand is a velocity, and how does that differ from total distance traveled?
- How would you explain to a friend why accumulating a rate of change recovers the total change?
Build the foundations first
Integrals & the fundamental theorem of calculus builds on these concepts. If any feel shaky, start there.