Using integration, find the area of the region in the first quadrant enclosed by the x-axis, the line y=x and the circle x² + y² = 32.

Equation of the circle given is x² + y² = 32
Radius of the given circle is = √32 = 4√2
x² + y² = 32 ⇒ y = √(32 - x²)
(4,4) is the point of Intersection of the circle and the line y = x
This can be calculated by solving the equations y = x and x² + y² = 32 simultaneously
The required area is the area of region OAB.
A = (4√2, 0), B = (4, 4) and O = (0, 0)
Area = ∫₄₀ x dx + ∫₄√₂⁴ √(32 - x²) dx = (x²/2)₄₀ + (x√(32 - x²)/2 + 32/2 sin⁻¹(x/4√2))₄√₂⁴
= 8 + 0 + 16π/2 - 4π = 4π
This is the required answer.