Definite Integrals & Series

To find the area under a curve, the first move is to slice the interval into many thin rectangles and add them all up. The more rectangles you use, the tighter the staircase hugs the curve, and pushing their number to infinity gives exactly the definite integral. That is why swapping the k/n inside the sum for x and the 1/n in front for dx turns a messy limit into one integral. Drag the number of rectangles n below and watch their total area close in on the integral value 1/3.

Slice Thin, Sum Up, Get Area

Number of rectangles n6

👀 See It

①Split the interval into n parts and add all rectangle areas

②As n grows the staircase hugs the curve

③In the limit, the sum is exactly the definite integral (area) — this is the Riemann sum

The Riemann Sum

The heart of the Riemann sum

lim_n→∞ 1n ∑_k=1ⁿ f(kn) = ∫₀¹ f(x) dx

1/n is the width (Δx), k/n is the position (x) — the limit of the sum is the integral

General interval [a,b]

lim_n→∞ ∑_k=1ⁿ f(a + (b−a)kn)·b−an = ∫_a^b f(x) dx

Rectangles of width (b−a)/n at position a+(b−a)k/n

Turning the Limit Sum Into an Integral

🔁 Substitution rule

①Replace k/n with x

②Replace 1/n with dx

③The sum over k=1..n becomes the integral from 0 to 1

④Even messy limit sums collapse to one integral by this rule

Compute It Directly

Example 1

Find lim_n→∞ 1n ∑_k=1ⁿ kn.

Replace k/n with x and 1/n with dx.

= ∫₀¹ x dx

Evaluate the integral.

= [x²2]₀¹ = 12

▸ 1/2

See the k/n inside as the variable x and the leading 1/n as dx — done.

Example 2

Find lim_n→∞ 1n ∑_k=1ⁿ (kn)².

Convert to an integral by the same rule.

= ∫₀¹ x² dx

Evaluating gives

= [x³3]₀¹ = 13

▸ 1/3

(k/n)² is x², 1/n is dx — powers do not change the rule.

Wrap-up

Key result

lim_n→∞ 1n ∑_k=1ⁿ f(kn) = ∫₀¹ f(x) dx

The sum of infinitely many rectangles = the definite integral (area)

2020 CSAT Math (Calculus) type, adapted

Find lim_n→∞ 1n ∑_k=1ⁿ √(k/n).

①1/2

②2/3

③3/4

④1

⑤Diverges

▸ ② 2/3

Replacing k/n→x and 1/n→dx gives an integral.

= ∫₀¹ √x dx

Integrate x^1/2.

= [23 x^3/2]₀¹ = 23

🎯 Exam Points

①Seeing 1/n·Σf(k/n) means ∫₀¹f(x)dx at once

②The k/n→x, 1/n→dx substitution is key

③Roots and powers do not change the rule

④General interval: width (b−a)/n, position a+(b−a)k/n

⑤If the 1/n in front is missing, it is not a Riemann sum

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Functions Defined by Integrals

Area Between Curves

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