Lecture #4: Applications of the Integral: Areas & Arc Lengths

Geometric Meaning of the Riemann Sum?

What does

\[ \int_a^b f(x) \, dx = \lim_{\Delta x \to 0} \sum_{i=1}^n f(x_i^*) \Delta x \]

actually mean?

The Integral as the Area Between Functions

If \(f(x) \ge g(x)\) everywhere on an interval \([a, b]\), then \(\displaystyle\int_a^b f(x) - g(x) \, dx\) is the area between the graphs of the function.

Example: The Integral as the Area Between Functions

Example: Find the area of the region bounded by the graphs of the functions \(y = x + 4\) and \(y = 3 - \dfrac{x}{2}\), from \(x = 1\) to \(x = 5\).

Example: The Integral as the Area Between Functions

Example: Find the area of the region bounded by the graphs of the functions \(y = x^3 - x\) and \(y = x\).

Example: The Integral as the Area Between Functions

Example: Find the area of the region bounded by the graphs of the functions \(y = x^2\) and \(y = 2 - x\).

Example: The Integral as the Area Between Functions

Example: Find the area of the region bounded by the graphs of the functions \(x = \dfrac{(y - 2)^2}{3}\) and \(y = 8 - x\).

The Integral as Arc Length

Given a curve, can you find its length?

Mean Value Theorem

If \(f(x)\) is continuous on \([a, b]\) and differentiable on \((a, b)\), then there exists a point \(c\) such that \(a \le c \le b\) and:

\[ f'(c) = \frac{f(b) - f(a)}{b - a}. \]

Mean Value Theorem

The Integral as Arc Length

If \(f(x)\) is a differentiable function on \([a, b]\), then the arc length of the graph of \(f(x)\) over \([a, b]\) is

\[ \int_a^b \sqrt{1 + [f'(x)]^2} \, dx. \]

Example: The Integral as Arc Length

Example: Let \(f(x) = 2x^{\frac{3}{2}}\). Calculate the arc length of the graph of f(x) over the interval \([0, 1]\).

Example: The Integral as Arc Length

Example: Find the circumference of a circle of radius \(a\).

\[ x^2 + y^2 = a^2 \]