Definite Integration — Class 12 IMO

Definite Integral and the FTC

The definite integral $\displaystyle\int_a^b f(x)\,dx$ is a number — the net signed area between the curve and the $x$-axis from $a$ to $b$.

Fundamental Theorem of Calculus

If $F$ is an antiderivative of $f$, then

$$\int_a^b f(x)\,dx=F(b)-F(a).$$

So you integrate, then substitute the limits and subtract. The constant $C$ cancels, which is why definite integrals have no $+C$.

Useful properties

$\displaystyle\int_a^b f\,dx=-\int_b^a f\,dx$, and $\displaystyle\int_a^a f\,dx=0$.
$\displaystyle\int_a^b f\,dx=\int_a^c f\,dx+\int_c^b f\,dx$.
$\displaystyle\int_0^a f(x)\,dx=\int_0^a f(a-x)\,dx$ (the "king" property).
$\displaystyle\int_{-a}^{a} f(x)\,dx=2\int_0^a f\,dx$ if $f$ is even, and $0$ if $f$ is odd.

Example 1: Evaluate $\displaystyle\int_0^2 x^2\,dx$.

Antiderivative $\tfrac{x^3}{3}$. So $\left[\tfrac{x^3}{3}\right]_0^2=\tfrac{8}{3}-0=\dfrac{8}{3}.$

Example 2: Evaluate $\displaystyle\int_0^{\pi} \sin x\,dx$.

$[-\cos x]_0^{\pi}=-\cos\pi-(-\cos0)=-(-1)+1=2.$

Example 3: Evaluate $\displaystyle\int_{-1}^{1} x^3\,dx$.

$x^3$ is an odd function and the limits are symmetric, so by the odd-function property the integral is $0$.

Example 4: Evaluate $\displaystyle\int_1^3 (2x+1)\,dx$.

Antiderivative $x^2+x$. $[x^2+x]_1^3=(9+3)-(1+1)=12-2=10.$

Quick recap

$\int_a^b f\,dx=F(b)-F(a)$ (Fundamental Theorem); no $+C$ for definite integrals.
Swapping limits flips the sign; equal limits give $0$.
King property: $\int_0^a f(x)\,dx=\int_0^a f(a-x)\,dx$.
Symmetric limits: even $\Rightarrow 2\int_0^a$, odd $\Rightarrow 0$.

✓ Quick check

Evaluate ∫₋₂² |x| dx.

|x| is even, so ∫₋₂² |x| dx = 2 ∫₀² x dx = 2 · [x²/2]₀² = 2 · 2 = 4.

If f(x) is continuous and ∫₀ˣ f(t) dt = x² + 2x, then f(2) equals:

By FTC, differentiating both sides: f(x) = 2x + 2. So f(2) = 4 + 2 = 6.

Properties of Definite Integrals

Key properties: ∫[a,b] f = −∫[b,a] f; ∫[a,b] f = ∫[a,c] f + ∫[c,b] f; and ∫[0,a] f(x)dx = ∫[0,a] f(a−x)dx.

Symmetry: ∫[−a,a] f = 0 if f is odd, and 2∫[0,a] f if f is even.

Example 1: Evaluate ∫[−2,2] x³ dx.

0 — the integrand is odd.

Example 2: ∫[−1,1] x² dx using symmetry?

2∫[0,1] x² dx = 2/3.

Quick recap

Odd function over [−a, a] integrates to 0.
∫[0,a] f(x)dx = ∫[0,a] f(a−x)dx (king property).

✓ Quick check

The value of ∫₀² x² [x] dx, where [x] is greatest integer function, is:

∫₀² x² [x] dx = ∫₀¹ x²·0 dx + ∫₁² x²·1 dx = 0 + [x³/3]₁² = 8/3 − 1/3 = 7/3. We check: x in [0,1): [x]=0. x in [1,2): [x]=1. At x=2, [2]=2 but point measure zero. So ∫ = 0 + ∫₁² x² dx = [x³/3]₁² = 8/3 − 1/3 = 7/3. Option 3 is 7/3. So correct index 3.

The area bounded by y = sin 2x, x = 0, x = π/2 and the x-axis is:

Area = ∫₀^(π/2) |sin 2x| dx. sin 2x ≥ 0 in [0, π/2], so area = ∫₀^(π/2) sin 2x dx = [−1/2 cos 2x]₀^(π/2) = −1/2(−1 − 1) = 1.

Integral as a Limit of a Sum

The definite integral is the limit of a Riemann sum: ∫[a,b] f(x)dx = lim (b−a)/n · Σ f(a + r(b−a)/n) as n → ∞.

This first-principles view links integration to area accumulation.

Example 1: What does the Riemann sum approximate?

The area under the curve between a and b.

Example 2: As n → ∞, the sum tends to?

The exact definite integral.

Quick recap

∫ = limit of Riemann sums as n → ∞.
It represents accumulated signed area.

✓ Quick check

The area enclosed by y = √x, x = 1, x = 4 and the x-axis is:

Area = ∫₁⁴ x^(1/2) dx = [ (2/3) x^(3/2) ]₁⁴ = (2/3)(8 − 1) = 14/3.

A steel plant in Bhilai has a production rate given by P'(t) = 50 + 4t tonnes/hour. What is the total steel produced in the first 5 hours?

Total production = ∫[0 to 5] (50 + 4t) dt = [50t + 2t²] from 0 to 5 = 250 + 2(25) = 250 + 50 = 300 tonnes.