Given the function $$ f(x)= an x $$ for $$ x $$ between $$ -\frac{\pi}{4} $$ and $$ \frac{\pi}{4} $$, find: The net area $$ =0 $$ The total area $$ =[?] $$ Round to the nearest thousandth.

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Given the function $$ f(x)=	an x $$ for $$ x $$ between $$ -\frac{\pi}{4} $$ and $$ \frac{\pi}{4} $$, find: The net area $$ =0 $$ The total area $$ =[?] $$ Round to the nearest thousandth.

UpStudy AI · Accepted Answer

1. The net area is given by the integral
   $$
   \int_{-\frac{\pi}{4}}^{\frac{\pi}{4}} \tan x\, dx.
   $$
   Since $$\tan x$$ is an odd function and the interval is symmetric about 0, the net area is
   $$
   0.
   $$

2. The total area is the integral of the absolute value
   $$
   \int_{-\frac{\pi}{4}}^{\frac{\pi}{4}} |\tan x|\, dx.
   $$
   Because $$\tan x$$ is odd and its absolute value is even, we can write
   $$
   \int_{-\frac{\pi}{4}}^{\frac{\pi}{4}} |\tan x|\, dx = 2\int_{0}^{\frac{\pi}{4}} \tan x\, dx.
   $$

3. To evaluate the integral, recall that
   $$
   \int \tan x\, dx = -\ln|\cos x| + C.
   $$
   Hence, for the interval from 0 to $$\frac{\pi}{4}$$:
   $$
   \int_{0}^{\frac{\pi}{4}} \tan x\, dx = \left[-\ln|\cos x|\right]_{0}^{\frac{\pi}{4}} = -\ln\left|\cos \frac{\pi}{4}\right| + \ln|\cos 0|.
   $$
   Since $$\cos \frac{\pi}{4} = \frac{\sqrt{2}}{2}$$ and $$\cos 0 = 1$$, this becomes
   $$
   -\ln\left(\frac{\sqrt{2}}{2}\right) + \ln(1) = -\ln\left(\frac{\sqrt{2}}{2}\right).
   $$
   
4. Multiply this result by 2 to get the total area:
   $$
   2\left[-\ln\left(\frac{\sqrt{2}}{2}\right)\right] = -2\ln\left(\frac{\sqrt{2}}{2}\right).
   $$
   Notice that
   $$
   \frac{\sqrt{2}}{2} = \frac{1}{\sqrt{2}},
   $$
   so we have
   $$
   -2\ln\left(\frac{1}{\sqrt{2}}\right) = 2\ln(\sqrt{2}).
   $$
   Since $$\ln(\sqrt{2}) = \frac{1}{2}\ln2$$, it follows that
   $$
   2\ln(\sqrt{2}) = 2\left(\frac{1}{2}\ln2\right) = \ln2.
   $$

5. Rounding to the nearest thousandth, we have
   $$
   \ln2 \approx 0.693.
   $$

The total area is $$ \ln2 \approx 0.693 $$.
The total area is approximately 0.693.

Given the function for
between and , find:
The net area
The total area
Round to the nearest thousandth.

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Given the function for between and , find: The net area The total area Round to the nearest thousandth.