1) The <em>net</em> area between the two functions is 2.
2) The <em>net</em> area between the two functions is 4/3.
3) The <em>net</em> area between the two functions is 17/6.
4) The <em>net</em> area between the two functions is approximately 1.218.
5) The <em>net</em> area between the two functions is 1/2.
<h3>How to determine the area between two functions by definite integrals</h3>
The area between the two curves is determined by <em>definite</em> integrals for a interval between two values of <em>x</em>. A general formula for the <em>definite</em> integral is presented below:
(1)
Where:
- <em>a</em> - Lower limit
- <em>b</em> - Upper limit
- <em>f(x)</em> - "Upper" function
- <em>g(x)</em> - "Lower" function
Now we proceed to solve each integral:
<h3>Case I -
and
</h3>
The <em>lower</em> and <em>upper</em> limits between the two functions are 0 and 1, respectively. The definite integral is described below:
The <em>net</em> area between the two functions is 2.
<h3>Case II -
and
</h3>
The lower and upper limits between the two functions are -3 and -1, respectively. The definite integral is described below:
The <em>net</em> area between the two functions is 4/3.
<h3>Case III -
and
</h3>
The definite integral is described below:
The <em>net</em> area between the two functions is 17/6.
<h3>Case IV -
and
</h3>
The definite integral is described below:
The <em>net</em> area between the two functions is approximately 1.218.
<h3>Case V -
and
</h3>
This case requires a combination of definite integrals, as <em>f(x)</em> may be higher that <em>g(x)</em> in some subintervals. The combination of definite integrals is:
The <em>net</em> area between the two functions is 1/2.
To learn more on definite integrals, we kindly invite to check this verified question: brainly.com/question/14279102