Approximate the arc length of y = cos x + 5 sin x from x = 0 to x = $\pi$.
A) 118.418
B) 10.882
C) 21.764
D) 5.441
E) 10.682

Question

Approximate the arc length of y = cos x + 5 sin x from x = 0 to x =  $\pi$.
A) 118.418
B) 10.882
C) 21.764
D) 5.441
E) 10.682

Quizplus · Accepted Answer

The arc length $L$ of a function $y = f(x)$ from $x = a$ to $x = b$ can be found using the formula $L = \int_a^b \sqrt{1 + [f'(x)]^2} dx$. For $y = \cos(x) + 5\sin(x)$, the derivative $y' = -\sin(x) + 5\cos(x)$. Squaring and adding 1 gives $1 + (-\sin(x) + 5\cos(x))^2 = 1 + \sin^2(x) - 10\sin(x)\cos(x) + 25\cos^2(x)$. Simplifying using trigonometric identities, we get a constant under the square root, which simplifies the integral to a basic form. The exact integral calculation might be complex, but when performed correctly, it yields an approximate value of $10.882$, making option B the correct choice.

Approximate the Arc Length of Y = Cos X π\piπ A) 118

Approximate the Arc Length of Y = Cos X $\pi$
A) 118