Find the area of region

Question

Example 1: let F(x,y) = (xy2,y+x).verify greens theorem for the region bounded by

Y = x², y=x, (x,y≥0).

Example 2: redo example 2 in the line integrals.evaluate

c∫ x²dx + xy dy

Over the unit square in the diagram below.

Example 3: let f(x,y) = (2y+e^x, x+sin(y²))

Evaluate c∫ f.ds where c is the unit circle centred at(0,0) traversed anticlockwise.

.calculation of line integral is too hard ,so use greens theorem.

Example  4: evaluate ∫ f.ds where

F (x,y) = (-y/x²+y²,x/x²+y²)

And c is the curve 3x²+y² = 1 oriented anticlockwise.

Area of region in xy plane

If c is a simple closed curve that bounds a region D ,than

Area of D = ½ ∫_c=∂d  xdx-ydx

Example 5 : find the area of region enclosed by