Find the area that contains all points

Question

x² + y² ≤ 4 and x + y ≥ 2

Find the area that contains all points which coordinates satisfy the inequalities

A. 3π + 2

B. 3π + 4

C. π - 2

D. π + 4

Answer 1

Solving these functions for y in standard form

f(x) = sqrt(4-x^2)

g(x) = 2-x

They intersect at (0,2) and (2,0) with shading above the line but below the circle.

The integral ( f(x)-g(x)) = x* sqrt( 1- x^2/4) - x^2/2 + 2*arcsin(x/2) + C

[ trig substitution x = sin t needed to integrate f(x) ]

For x= 0, The entire function vanishes ...

For x=2, the definite integral, is pi-2 because the sqrt term vanishes, the second term is -2,

and 2*arcsin(1) = 2 * pi/2 = pi

So the definite integral is pi -2