Question

Find the area of the region bounded by y=1/x^2,y=4, and x=5. Use dy to differentiate and/or integrate.

Answer 1

Explanation:

Let
`f(x) = 4` and
`g(x) = (1)/(x^2)`. The area A of the region bounded by the given lines is

`\displaystyle A = \int [f(x) - g(x)]dx`

Note that
`g(x) = (1)/(x^2)` intersects y = 4 at x = 1/2 so the limits of integration go from x = 1/2 to x = 5. The area integral can then be written as

`\displaystyle A = \int_{(1)/(2)}^(5)\left(4 - (1)/(x^2)\right)dx`

`\:\:\:\:= \left(4x + (1)/(x)\right)_{(1)/(2)}^5`

`\:\:\:\:= (20 + (1)/(5)) - (2 + 2) = (81)/(5) = 16(1)/(5)`