# Chapter 16: Definite Integrals

> Area as a limit: the definition of ∫ₐᵇ f(x)dx via Riemann sums, careful properties and sign conventions, true areas with absolute values, area between curves, upper and lower estimates, and the average value theorem.

**Canonical:** https://calculus.kitzo.work/chapter/definite-integrals  
**Level:** Core  
**Estimated time:** 55 minutes  
**Exercises:** 8 (auto-marked; answers stay server-side)

## Topics
- Riemann sums
- Definition of ∫ₐᵇ f(x)dx
- Properties of integrals
- Area between curves
- Upper & lower estimates
- Average value theorem
- Mean Value Theorem for integrals

**Why this matters**

Integration adds up infinitely many infinitely thin slices. Master that one move and area under curves, accumulated change, total distance from a speed profile all collapse into a single limit.

## 1 · From rectangles to a limit

The basic idea is embarrassingly simple. To find the area between a curve and the x-axis, chop the region into many skinny vertical strips, approximate each strip by a rectangle, add up their areas, and then let the strips get infinitely thin. The sum marches to a single, well-defined number — the **definite integral**.

One subtlety before we start: the integral does not always count area as a positive quantity. If the curve dips below the x-axis, those strips contribute *negative* area. The integral measures **net signed area** — area above the axis minus area below it. Keeping that in mind now prevents a world of pain later.

## 2 · The definition, formally

Chop [a, b] into n equal subintervals of width Δx = (b − a)/n, with sample points xᵢ* = a + iΔx (left, right — the limit does not care). The **Riemann sum** is Σᵢ₌₁ⁿ f(xᵢ*) Δx:

`∫ₐᵇ f(x)dx = lim (n→∞) Σᵢ₌₁ⁿ f(xᵢ*) Δx, where Δx = (b−a)/n`

If this limit exists and is the same for every sample choice, f is **integrable** on [a, b]. Every continuous function is integrable; some discontinuous ones are too (jumps are fine).

Two summation formulas do the heavy lifting in honest-to-goodness computations:

  - Σᵢ₌₁ⁿ i = n(n+1)/2

  - Σᵢ₌₁ⁿ i² = n(n+1)(2n+1)/6

**Worked example**

**Example 1 — Compute ∫₀ᵃ x² dx straight from the definition.**

Step 1: Δx = a/n and take right endpoints xᵢ = ia/n.

Step 2: Σ (ia/n)² · (a/n) = (a³/n³) Σ i² = (a³/n³) · n(n+1)(2n+1)/6.

Step 3: Expand: (a³/6) · (n(n+1)(2n+1))/n³ = (a³/6) · (1 + 1/n)(2 + 1/n).

Step 4: Let n → ∞. The 1/n terms die, leaving (a³/6)·2 = **a³/3**.

Sanity check: ∫₀² x² dx = 8/3. This is the only way to evaluate an integral before the Fundamental Theorem arrives next chapter — and next chapter you will laugh at Step 1.

## 3 · Properties you will use constantly

  - **Additivity:** ∫ₐᵇ f + ∫ᵇᶜ f = ∫ₐᶜ f — integrals stitch end to end.

  - **Swapped limits:** ∫ᵇₐ f(x)dx = −∫ₐᵇ f(x)dx. Flip the limits, flip the sign. Consequently ∫ₐᵃ f = 0.

  - **Constant multiple:** ∫ₐᵇ c·f(x)dx = c·∫ₐᵇ f(x)dx.

  - **Sum:** ∫ₐᵇ (f + g)dx = ∫ₐᵇ f dx + ∫ₐᵇ g dx. (But never split an integral of a product or quotient — only sums.)

  - **Comparison:** if f(x) ≤ g(x) on [a, b], then ∫ₐᵇ f ≤ ∫ₐᵇ g.

## 4 · Unsigned areas and absolute values

The symbol ∫ₐᵇ f dx always returns signed area. For the *actual* area, split the interval wherever f crosses zero and integrate |f| piecewise:

`Area = ∫ₐᶜ f dx − ∫ᶜᵇ f dx  (f ≥ 0 on [a, c], f ≤ 0 on [c, b])`

**Area between two curves:** on [a, b] with f(x) ≥ g(x), the area is ∫ₐᵇ [f(x) − g(x)]dx — the top curve minus the bottom curve. It does not matter which one dips below the axis; only which one is on top matters.

**Area between a curve and the y-axis:** horizontal slices are often easier. If x is a function of y, area = ∫_{c}^{d} x(y) dy — or right-curve minus left-curve when two curves face the y-axis.

**Worked example**

**Example 2 — Area between y = x and y = x² on [0, 1].**

Step 1: which curve is on top? On (0, 1), x ≥ x² — the line wins.

Step 2: set up A = ∫₀¹ (x − x²)dx.

Step 3: using ∫₀¹ x dx = 1/2 (triangle) and ∫₀¹ x² dx = 1/3 (from Example 1):

A = 1/2 − 1/3 = **1/6**.

## 5 · Upper and lower estimates

For a monotone (increasing) function, left rectangles undershoot and right rectangles overshoot:

`Lₙ ≤ ∫ₐᵇ f ≤ Uₙ`

Lₙ uses left endpoints, Uₙ right endpoints (swap for decreasing f). The gap Uₙ − Lₙ shrinks like (b−a)·(max|f′|)/n. This is how computers approximate integrals — and a fine sanity-check.

## 6 · The average value theorem

The **average value** of f on [a, b] is the height of the rectangle with the same width and area:

`f_avg = (1/(b−a)) ∫ₐᵇ f(x)dx`

The **Mean Value Theorem for integrals**: if f is continuous on [a, b], some c in [a, b] has f(c) = f_avg. A continuous function attains its own average. (The pathological f that is 1 on rationals and 0 on irrationals attains nothing and is not Riemann integrable at all.)

**Worked example**

**Example 3 — Average value of f(x) = x² on [0, 3].**

Step 1: ∫₀³ x² dx = 3³/3 = 9 (Example 1 pattern).

Step 2: divide by the width: f_avg = 9/3 = **3**.

Step 3: the MVT promises some c with c² = 3 — indeed c = √3 lies in [0, 3]. ✓

**Watch out**

The integral sign is a **signed** machine. ∫₀⁴ (x² − 4)dx is not the area between the parabola and the axis — the integrand is negative on [0, 2). Net value is −8/3; true area is 32/3. Split at the zero first, then take absolute values.

Chapter formula sheet
∫ₐᵇ f(x)dx = lim (n→∞) Σᵢ₌₁ⁿ f(xᵢ*) Δx,  Δx = (b−a)/n

Σᵢ₌₁ⁿ i = n(n+1)/2 ·  Σᵢ₌₁ⁿ i² = n(n+1)(2n+1)/6

∫ₐᵇ f + ∫ᵇᶜ f = ∫ₐᶜ f ·  ∫ᵇₐ f = −∫ₐᵇ f ·  ∫ₐᵇ cf = c ∫ₐᵇ f

∫ₐᵇ (f + g)dx = ∫ₐᵇ f dx + ∫ₐᵇ g dx ·  f ≤ g ⇒ ∫ₐᵇ f ≤ ∫ₐᵇ g

Area between curves: A = ∫ₐᵇ (top − bottom) dx

Unsigned area: split at zeros of f, integrate |f| piecewise

f_avg = (1/(b−a)) ∫ₐᵇ f(x)dx ·  MVT for integrals: ∃ c ∈ [a, b] with f(c) = f_avg

## Common mistakes

  - Forgetting that swapping limits flips the sign: ∫ᵇₐ = −∫ₐᵇ.

  - Reporting ∫ₐᵇ f as the area when f went negative — always split and use |f|.

  - Setting up top-minus-bottom backwards and getting a negative area.

  - Using left endpoints for the upper sum on an increasing function (that is the lower sum).

  - Applying the MVT for integrals to a discontinuous function — continuity is the hypothesis.

  - Believing every bounded function is integrable; wild oscillation everywhere defeats Riemann sums.

## Practice exercises

1. If f(x) dips below the x-axis on part of [a, b], then ∫ₐᵇ f(x)dx represents:

- A. the total area between f and the axis, always positive
- B. net signed area — area above the axis minus area below
- C. only the area trapped above the axis
- D. zero, because the curve crosses itself

2. Compute ∫₁³ 2x dx by viewing it as the area of a trapezoid.

3. If ∫₀² f(x)dx = 5 and ∫₂⁵ f(x)dx = 3, then ∫₀⁵ f(x)dx = ?

4. For f(x) = x² on [0, 2] with n = 2 subintervals, the UPPER Riemann sum (right endpoints, since f increases) equals?

5. If ∫₀³ f(x)dx = 7, then ∫₃₀ f(x)dx equals:

- A. 7
- B. −7
- C. 0
- D. 1/7

6. The average value of f(x) = 3x² on [0, 2] is?

7. Find the area between the curves y = 2x and y = x² on [0, 2]. Give your answer as a fraction.

8. The Mean Value Theorem for integrals guarantees that for a CONTINUOUS f on [a, b], there exists c in [a, b] such that:

- A. f(c) = 0
- B. f′(c) = f_avg
- C. f(c) equals the average value of f on [a, b]
- D. f(c) = f(a) + f(b)

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