What is the length of one cycle of the cardioid: #r = 5+5cos (theta)#?

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Answer 1 · 2016-04-27T14:58:50

Length of one cycle #=40" "#units

I consider this length of arc using a formula from Calculus with given polar curve equation

#r=5+5 cos theta#

#L=intds#

#ds=sqrt(r^2+((dr)/(d theta))^2) d theta#

For one cycle

#L=2*int_0^pi ds=2*int_0^pi sqrt(r^2+((dr)/(d theta))^2) d theta#

#L=2*int_0^pi sqrt((5+5 cos theta)^2+(-5 sin theta)^2) d theta#

#L=2*int_0^pi sqrt((25+50 cos theta+25 cos^2 theta+25 sin^2 theta)) d theta#

#L=2*int_0^pi sqrt((50+50 cos theta)) d theta#

#L=2sqrt50*int_0^pi sqrt((1+cos theta)) d theta#

#L=40" "#units

graph{(x^2+y^2-5x)^2-25(x^2+y^2)=0[-20,20,-10,10]}

God bless....I hope the explanation is useful.