Question

Find the area of the region bounded by y = x and circle x² + y² = 32 in the 1^st quadrant.

Answer 1

To find area of in first quadrant enclose by the circle

x² + y² = 32 …(i)

And y = x …(ii)

Solving these two equations, we get

Or 2x² = 32

Or x² = 16

Or x = \(\pm 4\)

\(\therefore\)  y = \(\pm 4\)

Equation (i) is a circle with centre (0, 0) and meets axes at A (±4√2, 0), (0, ±4√2). And y = x is a line passes through (0, 0) and intersect circle at B (4, 4).

These are shown in the graph below:

Region OABO = Region OCBO + Region CABC

The area of the region bounded by y = x and circle x² + y² = 32 is 4\(\pi\) sq. units.