Practice Maths

Fundamental Theorem of Calculus and Definite Integrals

Key Terms

A Riemann sum Σf(xi)δx approximates the area under a curve by summing rectangle areas
As δx → 0, the sum converges to the definite integral ∫ab f(x) dx
The FTC connects antidifferentiation to area: ∫ab f(x) dx = F(b) − F(a)
When f(x) ≥ 0 on [a,b], the definite integral equals the area under the curve
Notation: [F(x)]ab means F(b) − F(a)
Fundamental Theorem of Calculus
If F′(x) = f(x), then:
∫ab f(x) dx = [F(x)]ab = F(b) − F(a)

Riemann Sum (approximation):
Area ≈ Σi=1n f(xi) δx   where δx = (b − a)/n

Key properties:
∫ab f(x) dx = −∫ba f(x) dx    ∫aa f(x) dx = 0
∫ab kf(x) dx = k∫ab f(x) dx    ∫ac f(x)dx = ∫abf(x)dx + ∫bcf(x)dx
Worked Example 1: Evaluate ∫14 (3x² − 2x) dx

Step 1: Find the antiderivative: F(x) = x³ − x²
Step 2: Apply FTC:
∫14 (3x² − 2x) dx = [x³ − x²]14
= (4³ − 4²) − (1³ − 1²)
= (64 − 16) − (1 − 1)
= 48 − 0 = 48
Worked Example 2: Use a Riemann sum with n = 4 strips to estimate ∫02 x² dx, using right endpoints.

δx = (2−0)/4 = 0.5    x-values: 0.5, 1, 1.5, 2
Sum = 0.5[f(0.5) + f(1) + f(1.5) + f(2)]
= 0.5[0.25 + 1 + 2.25 + 4] = 0.5 × 7.5 = 3.75

Exact value: ∫02 x² dx = [x³/3]02 = 8/3 ≈ 2.667
The right-endpoint estimate (3.75) overestimates since f is increasing.
Hot Tip: Always include a constant of integration for indefinite integrals, but never for definite integrals — the constant cancels when you evaluate F(b) − F(a). Write square brackets [ ]ab to show you're about to substitute.

From Rectangles to Integrals: Building the Idea

The definite integral doesn't appear out of nowhere — it is the logical conclusion of a very natural question: how do we find the exact area under a curve?

Consider the area under y = f(x) from x = a to x = b. We can approximate it by dividing the interval into n equal strips of width δx = (b − a)/n. Each strip is approximated by a rectangle with height f(xi), giving area f(xi)δx. The total approximation is the Riemann sum:

Σi=1n f(xi)δx

As n → ∞ (strips get thinner and thinner), the rectangles perfectly fill the area, and the sum converges to the exact value — the definite integral ∫ab f(x) dx.

x y a b y = f(x) δx
As n increases, the rectangles better approximate the curve and the Riemann sum approaches the exact area.

The Fundamental Theorem of Calculus

The FTC is one of the most profound results in mathematics. It states that integration and differentiation are inverse operations. Specifically:

If F is any antiderivative of f (meaning F′(x) = f(x)), then:

∫ab f(x) dx = F(b) − F(a)

This is remarkable: to find an exact area, we just evaluate the antiderivative at two points and subtract. We don't need to sum infinitely many rectangles!

Why it works: The antiderivative F(x) accumulates area from some fixed reference point. F(b) is the total area from that reference to b; F(a) is the total area to a. Their difference is the area from a to b.

Evaluating Definite Integrals: Technique

The process has three clear steps:

  1. Find the antiderivative F(x) of the integrand f(x). Omit the +C (it cancels).
  2. Write [F(x)]ab using square bracket notation.
  3. Substitute upper limit b then lower limit a, and compute F(b) − F(a).

Example: ∫03 (2x + 1) dx = [x² + x]03 = (9 + 3) − (0 + 0) = 12

For functions involving ex: ∫01 e2x dx = [e2x/2]01 = e²/2 − 1/2 = (e²−1)/2

For trigonometric functions: ∫0π/2 cos x dx = [sin x]0π/2 = sin(π/2) − sin(0) = 1

Exam technique: Show the antiderivative in square brackets with limits clearly written, then substitute. Examiners award method marks for this step even if arithmetic errors follow.

Properties and Special Cases

These properties often simplify calculations:

  • Reversed limits: ∫ab f(x) dx = −∫ba f(x) dx. Swapping limits changes the sign.
  • Zero-width interval: ∫aa f(x) dx = 0. No interval, no area.
  • Constant multiple: ∫ab kf(x) dx = k∫ab f(x) dx
  • Sum/difference: ∫ab [f(x) ± g(x)] dx = ∫ab f(x) dx ± ∫ab g(x) dx
  • Splitting the interval: ∫ac f(x) dx = ∫ab f(x) dx + ∫bc f(x) dx (for any b between a and c)

Connection to area: When f(x) ≥ 0 on [a,b], the definite integral equals the geometric area under the curve. When f(x) < 0, the integral gives a negative value — this is a signed area, not a geometric area. This distinction is critical for area problems.

Riemann Sums: Approximation and Error

Riemann sums appear in the QCAA exam when you need to estimate an integral numerically or interpret the meaning of integration.

Left-endpoint rule: Use x0, x1, ..., xn-1 as sample points.

Right-endpoint rule: Use x1, x2, ..., xn as sample points.

Midpoint rule: Use midpoints of each subinterval — generally more accurate.

For an increasing function: right-endpoint overestimates, left-endpoint underestimates.

For a decreasing function: the opposite is true.

Accuracy improves as n increases (more strips, smaller δx).

Common error: Confusing the number of strips n with the number of x-values. With n strips, you have n+1 x-values (from x0 = a to xn = b).

Mastery Practice

See Answers ➔
  1. Evaluate ∫13 (4x − 1) dx.
  2. Evaluate ∫02 (x² + 3x) dx.
  3. Evaluate ∫1e (1/x) dx. Give an exact answer.
  4. Use a left-endpoint Riemann sum with n = 4 strips to estimate ∫04 x² dx. Show all x-values and rectangle heights used.
  5. Given that ∫15 f(x) dx = 7 and ∫35 f(x) dx = 2, find ∫13 f(x) dx.
  6. Evaluate ∫0π/4 2 sin x dx. Give an exact answer.
  7. Evaluate ∫02 e3x dx. Give an exact answer.
  8. A student writes ∫25 f(x) dx = −∫52 f(x) dx = −10. What is ∫25 f(x) dx? Explain why.
  9. Use a right-endpoint Riemann sum with n = 4 to estimate ∫13 (1/x) dx. Then evaluate the integral exactly and calculate the percentage error of your estimate.
  10. A function g satisfies g(1) = 3 and g(4) = 11. If g′(x) = f(x), explain what ∫14 f(x) dx represents and state its value. Include a sketch showing the relationship between F, f, and the definite integral.