Circle equation?

Question

Find the area of the region inside the circle x squared + y squared = 2 and above the line y=1.

Answer 1

x^2 + y^2 = 2
when y = 1, x = -1, +1
So we need to integrate (y-1)*dx from x = -1 to x = 1

ie integrate [sqrt(2 - x^2) - 1]dx

Make the sub x^2 = 2*sin^2 u to help integrate the first term.

See if you get pi/2 - 1

Answer 2

If x^2 + y^2 = 2, r = √2
The area sought is π/2 - 1 ≈ 0.5707963
(When y = 1, x = ± 1 The central angle between these two points is π/2. The area of the sector is the area of the segment minus the area of the triangle formed by the radii and the chord.)

Answer 3

first integrate the semicircle above the x-axis and find the area under the curve ie semi circle..let this be A1.

then drop a perpendicular from the point when the line y=1 cuts the circle to the x-axis.
there will be two such points..
so u'll get two rectangles and two triangles.
fin the area of the two rectangles and the two triangles..lets them be A2,A3.
hence required area A=A1-(A2+A3)

Answer 4

the answer is integral around the earth of f(x,y,z) for some function f times some constant k.