Question

Evaluate ∫∫ydxdy over the region
betwech the
parobola x²=y and the line x+y=2​

Answer 1

Step-by-step explanation:

The in tersection point of parabolas y = x2 and x = y2 is

y = ( v2)2 ⇒ y = y4

⇒y=0,y=1⇒x=0,x=1

So, the intersection point in 0(0, 0) and (1, 1).

∴ Required area = ∫10(y2−y1)dx

=∫10(x−−√−x2)dx

=[x3/23/2−x33]0=[23x3/2−x33]10

=[23(1)−13(1)+0−0]

=[23−13]=13

Alternate method

We know that, if parabolas are y2 = 4ax and x2 = 4, then area of bounded region is 4a⋅4b3

∴ Required area  =1×13=13