Question

in exercise 31-36 find a definite integral that represents the length of the specified polar curve, and then use graphing calculator or computer algebra system to approximate the value of integral

The entire limacon $r=1+2\sin \theta$

Short answer

The integral can be given as $L={\int }_0^{2\pi }\sqrt{\left(2\cos \theta \right)+(1+2\sin \theta )}d\theta$and arc length can be given as 13.37units

Step-by-step solution

Given information

The entire limacon$r=1+2\sin \theta$

Evaluate

We know that length of limacon can be given as

$$\begin{gathered} L={\int }_0^{2\pi }\sqrt{{(r\text{'})}^2+{\left(r\right)}^2}dr \\ Substitutingthevaluesweget \\ L={\int }_0^{2\pi }\sqrt{{\left(2\cos \theta \right)}^2+{(1+2\sin \theta )}^2}dr \\ wefirstsimplifythegivenexpression \\ weget, \\ L={\int }_0^{2\pi }\sqrt{4{\cos }^2\theta +1+4{\sin }^2\theta +4\sin \theta }d\theta \\ L={\int }_0^{2\pi }\sqrt{5+4\sin \theta }d\theta \\ Nowu\sin gCAScalculatorweget, \\ L=13.36units \end{gathered}$$