we have the polar equation
[tex]r=5sin(4\theta)[/tex]
using a graphing tool
The arc length is given by the formula
[tex]L=\int_a^b\sqrt{(r^{\prime}(\theta)^2+(r(\theta))^2}\text{ }d\theta[/tex]
where
Find out r' (a derivative of r)
[tex]r^{\prime}(\theta)=20cos(4\theta)[/tex]
a=0
b=pi/4
substitute given values in the formula
[tex]\begin{gathered} L=\int_0^{\frac{pi}{4}}\sqrt{(20cos(4\theta))^2+(5sin(4\theta))^2}\text{d}\theta \\ L=\int_0^{\frac{pi}{4}}\sqrt{400cos^2(4\theta)+25sin^2(4\theta)}\text{d}\theta \end{gathered}[/tex]
Solve the integral
The answer is
one minute, please
The arc length is L=10.723 units (three decimal places)
