Applications

The most important application of integration is differential equations. Integration is also used in geometry as alluded to in the beginning of this section. This is not limited to areas as seen but also to volumes as well as lengths of curves. For instance, the integral

L=\int_{a}^{b} \sqrt{1+\left(f^{\prime}(x)\right)^{2}} d x

gives the length of the curve $y=f(x)$ between $x=a$ and $x=b$ . To see where the formula comes from, consider a small segment of the curve $y=f(x)$ as shown in the diagram below:

small segment of the curve

Define $\Delta y=f(x+\Delta x)-f(x) \approx f^{\prime}(x) \Delta x$ . The length of the segment (joining the points $x$ and $x+\Delta x)$ is given by

\Delta l \approx \sqrt{(\Delta x)^{2}+\left(\Delta_{y}\right)^{2}} \approx \sqrt{1+\left(f^{\prime}(x)\right)^{2}} \Delta x

Summing up lengths of all small segments from $x=a$ to $x=b$ gives the integral shown at the start. An alternative formula for the length of a curve is:

L=\int_{t_{a}}^{t_{b}} \sqrt{(\dot{x}(t))^{2}+(\dot{y}(t))^{2}} d t

where the curve has the parametric form $x=x(t), y=y(t)$ for $t_{a} \leq t \leq t_{b}$ . Here, the dot notation represents differentiation wrt $t$ .

For example, say we wanted to calculate the arc length $S$ of the cycloid for $0 \leq t \leq 2 \pi$ . The equations of a cycloid created by a circle of radius 1 are given parametrically by,

x(t)=t-\sin t, \quad y(t)=1-\cos t .

To find the length we use the equation. Differentiating $x$ and $y$ wrt $t$ ,

\dot{x}=1-\cos t, \quad \dot{y}=\sin t

and plugging in the equation,

L=\int_{0}^{2 \pi} \sqrt{(1-\cos t)^{2}+(\sin t)^{2}} d t

leaves us to evaluate

L=\int_{0}^{2 \pi} \sqrt{2(1-\cos t)} d t

Another length formula is

L=\int_{\theta_{a}}^{\theta_{b}} \sqrt{(r(\theta))^{2}+\left(r^{\prime}(\theta)\right)^{2}} d \theta ;

which deals with a curve of the form $r=r(\theta)$ where $r$ and $\theta$ are polar coordinates. This formula can be used to compute the length of the cardioid defined by $r=1+\cos \theta$ .

Integration techniques Graphs