Question

Find the area of the region described.

The region between the line y=x and the curve y=2x√(25 - x^2) in the first quadrant. The total area of the shaded region is ___ (from 7087 to 100).

Answer 1

The area is
`(567)/(8)u^2`

Explanation:

The area of a flat region bounded by the graphs of two functions f (x) and g (x), with f (x)> g (x) can be found through the integral:

`\int\limits^b_a {[f(x) - g(x)]} \, dx`

The integration limits are given by the intersection points of the graphs of the functions in the first quadrant. Then, the cut points are:

`g(x) = x\\f(x) = 2x√(25-x^2)`

`x=2x√(25-x^2)\\x^2=4x^2(25-x^2)\\x^2(1-100+4x^2)=0\\x_1=0\\x_2=(3√(11))/(2)`

The area of the region is:

`\int\limits^b_a {[f(x) - g(x)]} \, dx = \int\limits^{(3√(11))/(2)}_0 {x(2√(25-x^2)-1)} \, dx = (567)/(8)u^2`