Question

Calculate the area under the graph of f(x) = 1/xlnx from x = e to x = e^e

Answer 1

Using integration, it is found that the area under the graph of f(x) in the desired interval is of 1 unit squared.

How is the area under a graph found?

The area under the graph of a curve f(x) between x = a and x = b is given by the following integral:

$A = \int_{a}^{b} f(x) dx$

In this problem, the function and the limits of integration are given by:

$f(x) = \frac{1}{x\ln{x}}$

$a = e, b = e^e$

Hence:

$A = \int_{e}^{e^e} \frac{1}{x\ln{x}} dx$

Using substitution:

$u = \ln{x}$

$du = \frac{1}{x} dx$

$dx = x du$

Hence:

$\int \frac{1}{x\ln{x}} dx = \int \frac{1}{u} du = \ln{u} = \ln{\ln{x}}$

Applying the Fundamental Theorem of Calculus:

$A = \ln{\ln{e^e}} - \ln{\ln{e}} = \ln{e\ln{e}} = \ln{\ln{e}} = \ln{e} - \ln{1} = 1$

Hence, the area under the graph of f(x) in the desired interval is of 1 unit squared.

To learn more about integration, you can take a look at brainly.com/question/20733870