Find the arc length of the graph of the function over the indicated interval. (Round your answer to three decimal places.)
y = ln (sin(x))
,[π/4,3π/4]
To find the arc length of the graph of the function y = ln(sin(x)) over the interval [π/4, 3π/4], we can use the formula for arc length:
L = ∫[a, b] √[1 + (f'(x))^2] dx
where a and b are the limits of the interval, and f'(x) is the derivative of the function with respect to x.
First, let's find the derivative of y = ln(sin(x)):
y' = (1 / sin(x)) * cos(x) = cos(x) / sin(x) = cot(x)
Next, we need to find the square root of 1 + (f'(x))^2:
√[1 + (cot(x))^2] = √[1 + cot^2(x)] = √[cosec^2(x)] = √[1 / sin^2(x)] = 1 / sin(x)
Now, we can set up the integral to find the arc length:
L = ∫[π/4, 3π/4] (1 / sin(x)) dx
To solve this integral, we can use a trigonometric identity to rewrite sin(x) in terms of cos(x):
L = ∫[π/4, 3π/4] (1 / sin(x)) dx
= ∫[π/4, 3π/4] (1 / √[1 - cos^2(x)]) dx
= ∫[π/4, 3π/4] (1 / √[1 - (1 - sin^2(x))]) dx
= ∫[π/4, 3π/4] (1 / √[sin^2(x)]) dx
= ∫[π/4, 3π/4] (1 / sin(x)) dx
Now, we can solve the integral:
L = ∫[π/4, 3π/4] (1 / sin(x)) dx
= ln|cosec(x) - cot(x)| + C (where C is the constant of integration)
Finally, we can evaluate the integral over the given interval:
L = ln|cosec(3π/4) - cot(3π/4)| - ln|cosec(π/4) - cot(π/4)|
Evaluating the natural logarithm of the trigonometric functions:
L = ln|√2 + 1| - ln|√2 - 1|
Using a calculator, we find:
L ≈ ln(2.414) - ln(0.414) ≈ 0.884
Therefore, the arc length of the graph of the function y = ln(sin(x)) over the interval [π/4, 3π/4] is approximately 0.884 (rounded to three decimal places).
To find the arc length of the graph of the function y = ln(sin(x)) over the interval [π/4, 3π/4], we will use the arc length formula for a function f(x) on [a, b]:
L = ∫ sqrt(1 + (f'(x))^2) dx
Step 1: Find the derivative of the function y = ln(sin(x)).
To find the derivative of y = ln(sin(x)), we can use the chain rule. The derivative of ln(u), where u = sin(x), is 1/u * du/dx. Therefore, the derivative of y = ln(sin(x)) is:
dy/dx = 1/sin(x) * cos(x)
Step 2: Square the derivative and add 1.
We need to square the derivative and add 1 to obtain the expression inside the square root in the arc length formula:
(1/sin(x) * cos(x))^2 + 1
Simplifying this expression, we get:
(1/sin^2(x)) * cos^2(x) + 1
Step 3: Integrate the expression from π/4 to 3π/4.
We now need to integrate the expression sqrt((1/sin^2(x)) * cos^2(x) + 1) over the interval [π/4, 3π/4].
Thus, we have:
L = ∫[(π/4 to 3π/4)] sqrt((1/sin^2(x)) * cos^2(x) + 1) dx
Evaluate this integral using numerical methods or a graphing calculator to find the arc length. Round your answer to three decimal places.
y = ln sinx
y' = 1/sinx * cosx = tanx
s = Int(sqrt(1+(y')^2)dx)[pi/4,3pi/4]
= Int(sqrt(1+tan^2(x))dx)[pi/4,3pi/4=
= Int(secx dx)[pi/4,3pi/4]
= ln|secx + tanx|[pi/4,3pi/4]
= ln|-1/√2 + 1| - ln|1/√2 + 1|
= ln|(1-√2/(1+√2)|
= ln|2√2-3|
= ln(3-2√2)