Question

Find the length of the graph of f(x)=ln(5sec(x)) for 0≤x≤π/6

Answer 1

The length of the graph of
`\(f(x) = \ln(5\sec(x))\)` for
`\(0 \leq x \leq (\pi)/(6)\)` is approximately 0.67.

Step-by-step explanation:

To find the length of the graph of f(x) over the given interval, we use the arc length formula for a curve y = g(x) on
`\([a, b]\)` :

`\[ L = \int_(a)^(b) √(1 + (g'(x))^2) \,dx \]`

In this case,
`\(g(x) = \ln(5\sec(x))\)` . We start by finding g'(x), the derivative of g(x). Using the chain rule and properties of trigonometric functions, we get:

`\[ g'(x) = (1)/(\cos(x)) \]`

Now, we substitute g'(x) into the arc length formula and integrate over the given interval
`\([0, (\pi)/(6)]\)`:

`\[ L = \int_(0)^{(\pi)/(6)} \sqrt{1 + (1)/(\cos^2(x))} \,dx \]`

This integral simplifies to:

`\[ L = \int_(0)^{(\pi)/(6)} \sec(x) \,dx \]`

Evaluating this integral, we find that the length L is approximately 0.67. Therefore, the length of the graph of f(x) for
`\(0 \leq x \leq (\pi)/(6)\)` is 0.67.

In summary, we used the arc length formula to calculate the length of the graph, derived the expression for g'(x), and performed the necessary integration to obtain the final result of approximately 0.67.