Finding the Area Between Two Curves: Graphical vs. Algebraic Methods

[stewe151]

I know how to do this graphically, but I can't remember how to set it up the long way. The equations are:

y^2=x and y=x-2

I know it should be easy, but it's late and I can't think...

 

[d_leet]

I know how to do this graphically, but I can't remember how to set it up the long way. The equations are:

y^2=x and y=x-2

I know it should be easy, but it's late and I can't think...

Find the intersection points then set up the integral from the one intersection point to the other of the "larger" curve minus the "smaller" curve.

 

[benorin]

Solve for the points of intersection of the two curves:

$$x=y^2$$

and $$y=x-2 \Rightarrow x=y+2$$

the intersection of these curves occurs at the values of y such that

$$y^2=y+2 \Rightarrow y^2-y-2=(y+1)(y-2)=0$$

so $$y=-1,2$$ and recall that $$x=y+2$$,

so the points are: (1,-1) & (4,2)

The integrand is easiest as: rightmost curve - leftmost curve = $$(y+2) - y^2$$

and the integral is then $$\int_{y=-1}^{2} (y+2-y^2)dy$$