Sketch the region bounded by the curves y=root of 5-x2 and y=|x-1| and find its area

given curves are y=5-x2 .......(1) and y=x-1.........(2)
curve (1) is the part of the circle x2+y2=5 which is above the x-axis
to draw the graph of y=x-1 , first we will draw the graph of y = x-1 and then we will take the image in x-axis of that part of the graph which is below x-axis.
from (1) and (2): 5-x2=x-1
5-x2=(x-1)25-x2=x2-2x+12x2-2x-4=0x2-x-2=0(x-2)(x+1)=0x=-1 , 2

Required area =-125-x2dx--12x-1 dx
-125-x2dx=12x5-x2+12*5sin-1x5|-12=1+52sin-125-{-1+52sin-1(-15)}=2+52[sin-125+sin-115]=2+52[sin-1{25*1-15+15*1-45}]=2+52*sin-1{25*25+15*15}=2+52*sin-1(1)=2+52*π2=5π4+2
-12x-1dx=-11(x-1)dx+12x-1dx=--11(x-1)dx+12(x-1)dx=x-x22-11+x22-x12=1-12-(-1-12)+42-2-(12-1)=12+32+12=52
thus the required area 
=5π4+2-52=5π4-12 

hope this helps you

  • 93
What are you looking for?