high school Definite Integral

Definite Integral

The area under a curve has no neat plug-in formula, so how can we find it? The idea is to pack the region with thin rectangles to estimate the area, then slice those rectangles thinner and thinner. As the rectangles get finer, the estimate creeps toward the true area, and that limit is exactly the definite integral. Push the n slider higher and the Riemann sum converges to the exact value as the error shrinks toward zero.

Slice the Area into Rectangles

📐 Core Idea

①Want the area under a curve → fill with rectangles

②Slice them thinner (n → ∞) → converges to the exact area

③This limit IS the definite integral!

From Riemann Sum to Integral

Partitions n5

Upper bound b2

Definition

∫_a^b f(x) dx = lim_n→∞ Σ_k=1ⁿ f(x_k) Δx

Limit of the partition method = area under the curve. Δx = (b − a)/n

🔑 Observations

①n = 2–5: thick rectangles, big error

②n = 20–50: rectangles fill the curve closely

③n → ∞: Riemann sum → integral (exact area)

Fundamental Theorem of Calculus

FTC

∫_a^b f(x) dx = F(b) - F(a) (with F'(x) = f(x))

Knowing an antiderivative lets you compute the integral directly — no partitions needed!

💡 Why FTC is Powerful

①Direct Riemann limits are complicated

②With FTC: find F(x), compute F(b) − F(a) — done

③Differentiation (slope) and integration (area) are inverses

④Split: ∫_a^c + ∫_c^b = ∫_a^b

⑤Even functions double on a symmetric interval; odd functions give 0

Worked Example

∫₀² x² dx = [x³3]₀² = 83 - 0 = 83

Find F(x) = x³/3, then evaluate F(2) − F(0)

Sign of the Integral vs Area

Upper bound b1.5

Area vs Integral

Area S = ∫_a^b |f(x)| dx (absolute value!)

Integral is signed; physical area uses |f(x)|

⚠ Watch Out

①Integral value ≠ area (parts below x-axis count as negative)

②For actual area, integrate |f(x)|

③Split intervals into positive/negative parts

Wrap-up

Core

∫_a^b f(x) dx = F(b) - F(a)

Fundamental Theorem of Calculus

🎯 Exam Points

①Integral = signed area — distinct from area!

②FTC: F(b) − F(a) — antiderivative difference

③Splitting: ∫_a^c + ∫_c^b = ∫_a^b

④Even function: ∫_{-a}^{a} f(x) dx = 2∫_0^a f(x) dx

⑤Odd function: ∫_{-a}^{a} f(x) dx = 0

Worked Examples & Past Exam

Example 1

Evaluate ∫₀² 3x² dx.

Find an antiderivative F(x) = x³.

∫ 3x² dx = x³

By the fundamental theorem, compute F(2) − F(0).

[x³]₀² = 2³ - 0³ = 8

▸ 8

A definite integral is F(b) − F(a) using an antiderivative (fundamental theorem of calculus).

Example 2

Evaluate ∫₁³ (2x − 1) dx.

Find an antiderivative F(x) = x² − x.

∫ (2x - 1) dx = x² - x

Compute F(3) − F(1).

[x² - x]₁³ = (9 - 3) - (1 - 1) = 6

▸ 6

Subtract the value at the lower limit from the value at the upper limit.

2022 KICE mock exam Math type, adapted

What is ∫₀¹ (3x² + 2) dx?

①3

②1

③5

④2

⑤4

▸ ① 3

Find an antiderivative F(x) = x³ + 2x.

∫ (3x² + 2) dx = x³ + 2x

Compute F(1) − F(0).

[x³ + 2x]₀¹ = (1 + 2) - 0 = 3

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Indefinite Integral

Applications of Definite Integrals

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