To solve for the area bounded by the functions [tex]\( f(x) = \tan x \)[/tex], [tex]\( g(x) = \cot x \)[/tex], and [tex]\( h(x) = \frac{4}{3 \pi} x - \frac{2}{3} \)[/tex] for the interval [tex]\( \frac{\pi}{4} \leq x \leq \frac{\pi}{2} \)[/tex], we need to consider the areas between each pair of functions and sum them up.
1. Understand the Interval and Functions:
- The functions [tex]\( f(x) = \tan x \)[/tex] and [tex]\( g(x) = \cot x \)[/tex] are trigonometric functions.
- [tex]\( h(x) = \frac{4}{3 \pi} x - \frac{2}{3} \)[/tex] is a linear function.
- We need to determine the areas where these functions intersect within the given bounds.
2. Intersection Points:
- For the given interval [tex]\( \frac{\pi}{4} \leq x \leq \frac{\pi}{2} \)[/tex], we consider both intersections and areas.
3. Calculate Area Between [tex]\( \tan x \)[/tex] and [tex]\( h(x) \)[/tex]:
- The upper bound [tex]\( \tan x \)[/tex] will be above [tex]\( h(x) \)[/tex] from [tex]\( \frac{\pi}{4} \)[/tex] to [tex]\( \frac{\pi}{2} \)[/tex].
[tex]\[ \text{Area}_1 = \int_{\frac{\pi}{4}}^{\frac{\pi}{2}} (\tan x - h(x)) \, dx \][/tex]
[tex]\[ \text{Area}_1 = \int_{\frac{\pi}{4}}^{\frac{\pi}{2}} \left( \tan x - \left( \frac{4}{3 \pi} x - \frac{2}{3} \right) \right) \, dx \][/tex]
This integral leads to an infinite area:
[tex]\[ \text{Area}_1 = \infty \][/tex]
4. Calculate Area Between [tex]\( h(x) \)[/tex] and [tex]\( \cot x \)[/tex]:
- The function [tex]\( h(x) \)[/tex] will be above [tex]\( \cot x \)[/tex] from [tex]\( \frac{\pi}{4} \)[/tex] to [tex]\( \frac{\pi}{2} \)[/tex].
[tex]\[ \text{Area}_2 = \int_{\frac{\pi}{4}}^{\frac{\pi}{2}} \left( h(x) - \cot x \right) \, dx \][/tex]
[tex]\[ \text{Area}_2 = \int_{\frac{\pi}{4}}^{\frac{\pi}{2}} \left( \frac{4}{3 \pi} x - \frac{2}{3} - \cot x \right) \, dx \][/tex]
This integral evaluates to:
[tex]\[ \text{Area}_2 = \log(\sqrt{2}/2) - \frac{0.0416666666666667 \pi}{1} = \log(\sqrt{2}/2) - 0.0416666666666667 \pi \][/tex]
5. Total Area:
- Add the areas calculated.
[tex]\[ \text{Total Area} = \text{Area}_1 + \text{Area}_2 \][/tex]
[tex]\[ \text{Total Area} = \infty + \left( \log(\sqrt{2}/2) - 0.0416666666666667 \pi \right) \][/tex]
[tex]\[ \text{Total Area} = \infty \][/tex]
Thus, the total area bounded by the given functions [tex]\( f(x) = \tan x \)[/tex], [tex]\( g(x) = \cot x \)[/tex], and [tex]\( h(x) = \frac{4}{3 \pi} x - \frac{2}{3} \)[/tex] over the interval [tex]\( \frac{\pi}{4} \leq x \leq \frac{\pi}{2} \)[/tex] is infinite.
