# Chapter 15: Introduction to Integration

> Sigma notation, telescoping sums, displacement versus distance, signed area, and the leap from rectangular approximation to the definite integral.

**Canonical:** https://calculus.kitzo.work/chapter/integration-intro  
**Level:** Core  
**Estimated time:** 45 minutes  
**Exercises:** 8 (auto-marked; answers stay server-side)

## Topics
- Sigma notation
- Telescoping sums
- Displacement vs distance
- Signed area
- Velocity from position
- Rectangular approximation

**Why this matters**

Everything in calculus until now ran from position to velocity to acceleration by differentiating. Integration is the reverse journey: rebuilding the accumulated total from the rate. This chapter builds the intuition for that reversal — adding slices, reading signs, and turning a sum into an area — so the definite integral arrives as an old friend rather than a new boss.

## 1 · Sigma notation: Σ

Σ is compact notation for 'add a pattern of terms'. Σ_{k=1}^{n} a_k means a_1 + a_2 + … + a_n, where k is the index and n the upper limit. Core rules worth memorizing:

- Σ_{k=1}^{n} c = n·c (constant terms)

- Σ_{k=1}^{n} (a_k ± b_k) = Σ a_k ± Σ b_k (linearity)

- Σ_{k=1}^{n} k·c = c·n(n + 1)/2 (constants factor out)

- Σ_{k=1}^{n} k = n(n + 1)/2 and Σ_{k=1}^{n} k² = n(n + 1)(2n + 1)/6

Sigma notation is not intellectual decoration — it is what the limit of a sum machinery needs in order to talk about 'infinitely many tiny pieces'.

## 2 · A nice telescoping example

**Worked example 1**

### Σ_{k=1}^{n} 1/(k(k + 1))

- **Decompose:** 1/(k(k + 1)) = 1/k − 1/(k + 1) (partial fractions — check by recombining).

- **Write terms:** (1/1 − 1/2) + (1/2 − 1/3) + (1/3 − 1/4) + … + (1/n − 1/(n + 1)).

- **Cancel neighbours:** every intermediate −1/k kills the next +1/k. Only 1 and −1/(n + 1) survive.

- **Result:** Σ_{k=1}^{n} 1/(k(k + 1)) = 1 − 1/(n + 1) = n/(n + 1). For n = 10: 10/11 ≈ 0.909.

Telescoping is the same trick behind product rules for sums: explode each term into a difference, and watch the middle collapse. Many definite-integral computations in later chapters rest on exactly this idea.

## 3 · Displacement vs distance travelled

Velocity is signed: driving forward at 60 km/h for 30 minutes covers +30 km. Reversing at 60 km/h for 15 minutes covers −15 km. **Displacement** is the net change in position: 30 − 15 = 15 km. **Distance travelled** ignores sign: 30 + 15 = 45 km. Whenever velocity changes sign, the two answers diverge — and the odometer, unlike the sat-nav, only counts the absolute value.

## 4 · Three cases of motion

- **Constant velocity v:** distance = v·t. Simple multiplication.

- **Constant acceleration a from rest:** v = a·t grows linearly, and distance = ½·a·t² — the average speed (a·t)/2 times t.

- **Varying velocity v(t) with a known formula:** no single multiplication works. Chop [0, T] into tiny intervals where v is nearly constant; each slice contributes ≈ v(t_i)·Δt; add them all up.

## 5 · Signed area vs unsigned area

**Worked example 2**

### Area between y = x and the x-axis on [−1, 2]

- **Below the axis** (x from −1 to 0): triangle with legs 1 → area ½, sitting under the axis, so its **signed** value is −½.

- **Above the axis** (x from 0 to 2): triangle with legs 2 → **signed** area +2.

- **Signed total:** −½ + 2 = 3/2. **Unsigned total (geometric area):** ½ + 2 = 5/2.

Signed area treats below-axis regions as negative — matching displacement. Unsigned area sums absolute values — matching odometer distance. Which one you want is a question about the problem, not about the curve.

## 6 · The journey: position from velocity

The fundamental loop of motion analysis: position s(t) → velocity v(t) = s′(t) → acceleration a(t) = v′(t). To run it backwards, accumulate the signed-area slices of v from a start time to t. A continuous velocity, approximated by rectangles on [0, T], gives

distance ≈ Σ v(t_i)·Δt, and displacement = lim (Σ v(t_i)·Δt) as the slices shrink

Two natural approximations sandwich the truth: **left-endpoint rectangles** (use the velocity at the left edge of each slice) underestimate when v is increasing, and **right-endpoint rectangles** overestimate in that case. Midpoint rectangles split the difference and usually land closest. The narrower the slices, the tighter the sandwich — squeezing Δt → 0 pins down one exact number: the definite integral.

Letting the slices shrink without bound defines the definite integral:

∫ from a to b of f(x)dx = lim n→∞ Σ_{i=1}^{n} f(x_i)·Δx, Δx = (b − a)/n

This is a preview, not yet a computational engine: the real machine (the Fundamental Theorem of Calculus — differentiate an antiderivative and subtract) arrives in the next chapter. For now: Σ is how we add; the integral sign ∫ is what the add becomes in the limit; f(x) is the height curve; dx is the slice width. Whenever you see ∫, picture an infinitely fine trilogy of rectangles nesting under the curve.

**Watch out**

Signed and unsigned answers differ exactly on stretches where f dips below the x-axis. If a problem asks 'total distance covered' and your velocity function goes negative, you must split the interval at the zero of v and add |area| on each piece — simply integrating once gives only the net displacement.

**Chapter formula sheet**

Σ_{k=1}^{n} k = n(n + 1)/2
Σ_{k=1}^{n} k² = n(n + 1)(2n + 1)/6
Σ_{k=1}^{n} (a_k ± b_k) = Σ a_k ± Σ b_k
Telescoping: 1/(k(k + 1)) = 1/k − 1/(k + 1)
displacement = Σ v·Δt (signed); distance = Σ |v|·Δt
∫_a^b f(x)dx = lim Σ f(x_i)·Δx
distance under a: from rest, s = ½at²

## Common mistakes

- Confusing displacement (signed) with distance travelled (unsigned) when velocity changes sign.

- Forgetting the correction −1/(n+1) in telescoping sums and reporting 1 instead of n/(n + 1).

- Using right-endpoint rectangles and expecting an underestimate for increasing v (they overestimate).

- Applying Σ rules to non-linear patterns like Σ 1/k² with the linear formula for Σ k.

- Reading ∫ as 'area' without splitting at zeros of f — it is signed area, and signs matter.

- Assuming Σ v(t_i)·Δt stops being approximate the moment any formula is available — only the limit Δt → 0 makes it exact.

## Practice exercises

1. Which closed form equals Σ_{k=1}^{n} k?

- A. n²
- B. n(n + 1)/2
- C. n(n − 1)/2
- D. (2n + 1)/2

2. Compute Σ_{k=1}^{100} k.

3. Compute Σ_{k=1}^{10} 1/(k(k + 1)). Give your answer as a decimal to 2 decimals.

4. A runner jogs 5 m/s for 60 s forward, then 5 m/s for 20 s backward. What is her total distance travelled?

- A. 200 m
- B. 300 m
- C. 400 m
- D. 350 m

5. Write the closed form for 1 + 2 + … + n using caret for powers if needed, e.g. n(n+1)/2.

6. An object released from rest accelerates at a = 10 m/s². How far does it travel in 3 s, in metres?

7. For f(x) = x on the interval [−1, 2], the signed area between the curve and the x-axis is:

- A. 5/2
- B. −3/2
- C. 3/2
- D. 2

8. Name a function whose derivative is 3x² (answer with a plus-c if you like), e.g. x^3.

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