The region enclosed by the given curves are integrated with respect to x and the area is 4.2724 square units.
In this question,
The curves are y = 2/x, y = 2/x^2, x = 4
The diagram below shows the region enclosed by the given curves.
From the diagram, the limit of x is from 1 to 4.
The given curves are integrated with respect to x and the area is calculated as
[tex]A=\int\limits^4_1 {\frac{2}{x} } \, dx -\int\limits^4_1 {\frac{2}{x^{2} } } \, dx[/tex]
⇒ [tex]A=2[\int\limits^4_1 {\frac{1}{x} } \, dx -\int\limits^4_1 {\frac{1}{x^{2} } } \, dx][/tex]
⇒ [tex]A=2[\int\limits^4_1( {\frac{1}{x} } - {\frac{1}{x^{2} } } )\, dx][/tex]
⇒ [tex]A=2[lnx-\frac{1}{x} ] \limits^4_1[/tex]
⇒ [tex]A=2[(ln4-\frac{1}{4} )-(ln1-\frac{1}{1} )][/tex]
⇒ A = 2[(1.3862-0.25) - (0-1)]
⇒ A = 2[1.1362 + 1]
⇒ A = 2[2.1362]
⇒ A = 4.2724 square units.
Hence we can conclude that the region enclosed by the given curves are integrated with respect to x and the area is 4.2724 square units.
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