Question

The length of curve r = e^θ, θ value ranges between 0 to π is________

(a) \(\sqrt{2} \)

(b) \(\sqrt{2} \,(e^2π – 1) \)

(c) \(\sqrt{2} \,(e^2π + 1) \)

(d) \(\sqrt{2} \,(e^2π) \)

I got this question by my college director while I was bunking the class.

My doubt is from Rectification in chapter Integral Calculus of Engineering Mathematics

Answer 1

Right answer is (b) \(\sqrt{2} \,(e^2π – 1) \)

The best explanation: \(s = \int_a^b \sqrt{r^2 + (\frac{dr}{dθ})^2} \,dθ \)

\(= \int_0^{2π} \sqrt{(e^θ)^2 + (e^θ)^2)} \,dθ \)

\(= \int_0^{2π} e^θ \sqrt{2} \,dθ \)

\(= (e^θ \sqrt{2}) |_0^{2π} \)

\(= \sqrt{2} (e^{2π}-1). \)