Quadratic Function Example in Integral Calculus

Intro

After the linear case, look at a slightly harder one: $y = x^2$.

Features:

• Parabola opening upward.
• On $[a,b]$ it forms a curvilinear trapezoid.
• No simple geometric formula—need integration.

Problem

Find the shaded area.

Integral setup

1. Split $[a,b]$ into thin strips.
2. Strip area = height × width.
3. Sum all strips.

For $y=x^2$:

• Height: $x_i^2$
• Width: $\Delta x$
• Area: $x_i^2 \Delta x$

Total area:

$\sum_{i=1}^{n} x_i^2 \cdot \Delta x$

Take the limit:

$\lim_{n \to \infty} \sum_{i=1}^{n} x_i^2 \cdot \Delta x$

Key formula

Sum of squares

$\sum_{i=1}^{n} i^2 = \frac{n(n+1)(2n+1)}{6}$

Proof uses induction/algebra; we accept it here.

Sketch of computation

With $x_i = a + i\Delta x$, $\Delta x = \tfrac{b-a}{n}$, expand the sum and apply the square-sum formula. Taking the limit gives the standard result:

$\int_a^b x^2 dx = \left[\frac{x^3}{3}\right]_a^b = \frac{b^3-a^3}{3}$

So the area under $y=x^2$ on $[a,b]$ equals $\dfrac{b^3-a^3}{3}$.

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