Approximate the arc length of x = sin 4y from y = 0 to y = $\pi$.
A) 4.406
B) 8.812
C) 17.624
D) 8.832
E) 8.712

Question

Approximate the arc length of x = sin 4y from y = 0 to y =  $\pi$.
A) 4.406
B) 8.812
C) 17.624
D) 8.832
E) 8.712

Quizplus · Accepted Answer

Using the formula for arc length,  $L=\int_{a}^{b}\sqrt{1+\left(\frac{dx}{dy}\right)^2} dy$, we have:
$$\frac{dx}{dy}=4\cos(4y) \implies \left(\frac{dx}{dy}\right)^2=16\cos^2(4y)$$
Thus, 
$$L=\int_{0}^{F1F1F1S1\pi}\sqrt{1+16\cos^2(4y)} dy=\frac{1}{4}\int_{0}^{4F1F1F1S1\pi}\sqrt{1+16\cos^2(u)} du,$$ where we made substitution $u=4y$.
This is a standard elliptic integral of the second kind, which does not have a closed form solution in terms of elementary functions. However, we can approximate it numerically to obtain:
$$L \approx \frac{1}{4}(8.812) = 2.203 \approx 2.2$$
Therefore, the best choice is B, 8.812 (rounded to three decimal places).

Approximate the Arc Length of X = Sin 4y from Y