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

Question

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

UpStudy AI · Accepted Answer

To find the net area and the total area under the curve of the function $$ f(x) = \tan x $$ between $$ -\frac{\pi}{4} $$ and $$ \frac{\pi}{4} $$, we will follow these steps:

1. **Net Area**: This is calculated by finding the definite integral of $$ f(x) $$ from $$ -\frac{\pi}{4} $$ to $$ \frac{\pi}{4} $$.
2. **Total Area**: This is calculated by taking the absolute value of the definite integral, which accounts for any areas below the x-axis.

### Step 1: Calculate the Net Area

The net area $$ A_{net} $$ can be calculated using the integral:

$$
A_{net} = \int_{-\frac{\pi}{4}}^{\frac{\pi}{4}} \tan x \, dx
$$

### Step 2: Calculate the Total Area

The total area $$ A_{total} $$ is given by:

$$
A_{total} = \left| A_{net} \right|
$$

Now, let's compute the integral for the net area.

First, we will find the integral of $$ \tan x $$. The integral of $$ \tan x $$ is:

$$
\int \tan x \, dx = -\ln |\cos x| + C
$$

Now, we will evaluate the definite integral from $$ -\frac{\pi}{4} $$ to $$ \frac{\pi}{4} $$.

Let's compute the definite integral:

$$
A_{net} = \left[ -\ln |\cos x| \right]_{-\frac{\pi}{4}}^{\frac{\pi}{4}}
$$

Now, I will calculate this integral.
Calculate the value by following steps:
- step0: Calculate:
  $$-\ln{\left(\left|\cos\left(\frac{\pi }{4}\right)\right|\right)}+\ln{\left(\left|\cos\left(-\frac{\pi }{4}\right)\right|\right)}$$
- step1: Calculate the absolute value:
  $$-\ln{\left(\frac{\sqrt{2}}{2}\right)}+\ln{\left(\left|\cos\left(-\frac{\pi }{4}\right)\right|\right)}$$
- step2: Calculate:
  $$-\ln{\left(\frac{\sqrt{2}}{2}\right)}+\ln{\left(\left|\cos\left(\frac{\pi }{4}\right)\right|\right)}$$
- step3: Calculate the absolute value:
  $$-\ln{\left(\frac{\sqrt{2}}{2}\right)}+\ln{\left(\frac{\sqrt{2}}{2}\right)}$$
- step4: Add the numbers:
  $$0\times \ln{\left(\frac{\sqrt{2}}{2}\right)}$$
- step5: Simplify:
  $$0\times \left(-\frac{1}{2}\ln{\left(2\right)}\right)$$
- step6: Calculate:
  $$0\times \left(-\frac{1}{2}\right)\times \ln{\left(2\right)}$$
- step7: Calculate:
  $$0$$
The net area $$ A_{net} $$ is calculated as follows:

$$
A_{net} = \left[ -\ln |\cos x| \right]_{-\frac{\pi}{4}}^{\frac{\pi}{4}} = -\ln |\cos(\frac{\pi}{4})| + \ln |\cos(-\frac{\pi}{4})| = 0
$$

### Step 3: Calculate the Total Area

Since the net area is $$ 0 $$, the total area $$ A_{total} $$ is:

$$
A_{total} = \left| A_{net} \right| = |0| = 0
$$

### Final Results:
- The net area $$ = 0 $$
- The total area $$ = 0 $$
The net area is 0, and the total area is 0.

Given the function for
between and , find:
The net area
The total area

Upstudy AI Solution

Answer

Solution

Step 1: Calculate the Net Area

Step 2: Calculate the Total Area

Step 3: Calculate the Total Area

Final Results:

Mind Expander

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Given the function for between and , find: The net area The total area