# Determine the length of the curve f (x) = sin (3 x) between x = 0 and x = 2.

## Question:

Determine the length of the curve {eq}f (x) = \sin (3 x) {/eq} between {eq}x = 0 {/eq} and {eq}x = 2 {/eq}.

## Definite Integral:

It measures the net area between the function and x-axis.

The constant, which was eliminated during the differentiation, has some definite value. That's the basic difference between the definite integral and indefinite integral.

Definite integrals can be used to find the area under, over, or between curves.

A general definite integral is taken in the complex plane, resulting in the contour integral.

Properties of definite integral:

1. The definite integral of a constant is equal to the length of the interval of integration.
2. The definite integral of a non-negative function is always greater than or equal to zero.
3. If the upper and lower limits of a definite integral are the same, the integral is zero.
4. Internal addition can be done in definite integral.

We know that for the given function y, the length of arc between {eq}x=a {/eq} to {eq}x=b {/eq} is L.

{eq}\displaystyle \begin{align} L &=\int_{a}^{b} f(x)dx\\ L &=\int_{0}^{2} \sin (3 x) dx\\ L &= -(0.3 \cos (3 x))_0^1\\ L &= -0.3(0.99 - 1)\\ L &= 0.003\ units \\\\ \end{align} {/eq}

Work as an Integral

from

Chapter 7 / Lesson 9
18K

After watching this video, you will be able to solve calculus problems involving work and explain how that relates to the area under a force-displacement graph. A short quiz will follow.