Area and the Definite Integral

This section covers the geometrical interpretation of integrals in determining areas under curves.

Definite Integrals and Areas Under Curves: Integrals calculate areas beneath curves, for instance, $\int_{1}^{4} (-x^2 + 4x + 4) dx$ for the function $f(x) = -x^2 + 4x + 4$.

Approximating Area with Rectangles: Areas can be approximated using rectangles. For $f(x) = 3x$ from $x = 0$ to $x = 20$, using more rectangles enhances estimation accuracy.

Riemann Sums: The area approximates via the Riemann Sum $\sum<em>{i=1}^{8} f(x</em>i)\Delta x$, obtaining exact results as $n \to \infty$.

Definition of the Definite Integral: For $[a, b]$ where f(x) > 0, the definite integral is defined as $\int<em>a^b f(x)dx = \lim</em>{n \to \infty} \sum<em>{i=1}^{n} f(x</em>i)\Delta x$. This integral measures the area between $x = a$, $x = b$, the x-axis, and $f(x)$.

Estimating Area: Several methods exist for estimating area: left-handed, right-handed, and midpoint methods.

Handling Areas Below the x-axis: Areas for functions below the x-axis necessitate taking the absolute value. For example, for $f(x) = -x^2 + 4x + 4$, the Riemann sum computes to $21.0625$, and for $f(x) = x^2 - 4x - 4$, the area is $|-21.0625| = 21.0625$.