How do you find the area of the region between the curves #y=x-1# and #y^2=2x+6# ?

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Answer 1 · 2014-09-20T22:49:58

The area of the region between #y=x-1# and #y^2=2x+6# is 18.

Let us look at some details.

The region looks like this:
enter image source here
By solving the equations for #x#,

#{(y=x-1 Leftrightarrow x=y+1),(y^2=2x+6 Leftrightarrow x=y^2/2-3):}#

Let us find he y-coordinates of the points of intersection.

#y^2=2(y+1)+6#

#Rightarrow y^2-2y-8=(y+2)(y-4)=0#

#Rightarrow y=-2,4#

So, the region spans from #y=-2# to #y=4#.

Now, we can find the area by

#A=int_{-2}^4[y+1-(y^2/2-3)]dy#

#=int_{-2}^4(4+y-y^2/2) dy#

#=[4y+y^2/2-y^3/6]_{-2}^4#

#=16+8-32/3-(-8+2+4/3)=18#