Using integration, it is found that the area of the shaded region is of [tex]240 - \ln{27}[/tex] units squared.
How is the area of a shaded region found?
- The area of shaded region, between two curves [tex]f(x)[/tex] and [tex]g(x)[/tex], considering [tex]f(x) > g(x)[/tex], between x = a and x = b, is given by the following integral:
[tex]A = \int_a^b f(x) - g(x) dx[/tex]
In this problem, the curves are:
[tex]f(x) = \sqrt[3]{x} = x^{\frac{1}{3}}[/tex]
[tex]g(x) = \frac{1}{x}[/tex]
The limits of integration are: [tex]a = 1, b = 27[/tex].
Hence:
[tex]A = \int_1^{27} \left(x^{\frac{1}{3}} - \frac{1}{x}\right) dx[/tex]
Applying the power properties of integration:
[tex]A = 3x^{\frac{4}{3}} - \ln{x}|_{x = 1}^{x = 27}[/tex]
Finally, applying the Fundamental Theorem of Calculus:
[tex]A = 3(27)^{\frac{4}{3}} - \ln{27} - 3(1)^{\frac{4}{3}} + \ln{1}[/tex]
[tex]A = 240 - \ln{27}[/tex]
The area of the shaded region is of [tex]240 - \ln{27}[/tex] units squared.
To learn more about integration, you can take a look at brainly.com/question/20733870
