Find the area of the region in the first quadrant enclosed by the y-axis, the line `y=x` and the circle `x^2+y^2=32 ,` using integration.

Explanation

Given, the curve is `x^2 + y^2 = 32` --- (1)
Given, the line is `y = x` --- (2)
We have to find the area of the region in the first quadrant enclosed by the x-axis, the line and the circle.

The standard form of equation of the circle is given by

`x^2 + y^2 = r^2` --- (3)
Comparing (1) and (3)

Radius of circle, `r = sqrt32 = 4sqrt2`
From (1) and (2),
`x^2 + x^2 = 322x^2 = 32

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