Calculus 2 Area of a Region

Question

I need this explained because I'm not really understanding it.

 

Sketch the region enclosed by the curves y=3|x| and y=x²−10.

 

List the points of intersection of these curves from LEFT to RIGHT in the form (x,y)(x,y) : ?

 

Due to the symmetry of the region with respect to the y-axis, the area A = 2 ∫ from 0 to a of h(x)dx where 

 

a= ?

h(x)= ?

After integrating, the area A= ?

 

 

Answer 1

 

A plot of the two functions is shown in:

 

https://www.wyzant.com/resources/files/496753/two_functions_plotted

 

 

To find where they intersect, you set them equal:

 

   3|x| = x²-10

 

So we can work without the absolute value operation, lets consider the two cases x<0 and x≥0 separately.  

 

For x<0:

       3(-x) = x²-10       (-x b/c we know x<0 and only want nonnegative values)

       x²+3x-10 = 0       (bring -3x over to right hand side, then rewrite eqn)

    (x+5)(x-2) = 0     

          x=-5  or x=2    (discard x>0 answer b/c we're considering negative x)

 

                             So x=-5 at one intersection

 

            To get the (x,y) coord's:  plug in x=-5 to either of your functions

 

For x≥0:

       3x = x²-10      

        x²-3x-10 = 0

       (x+2)(x-5) = 0

       x=-2 or x=5    (discard x<0 answer b/c we're considering x≥0 here)

 

                            x=5 is at the other intersection

 

           To get (x,y) coord's plug in x=5 to either function.

 

 

About integrating to find the area:

--------------------------------------

As you look at the two functions, notice that they are both symmetrical about the y-axis.  So, you can define h(x) to be:

 

h(x) = 3|x| - (x²-10).   Now, due to the symmetry, you can say the total area between the two curves is the same as twice the area under the part that goes to the right of the y-axis (i.e. the part where x≥0).  So,

 

   Area = 2 ∫ (3x - x² + 10) dx  evaluated from x=0 to x=5.   

 

We were able to just write 3x because we're integrating over the  x ≥ 0  portion only.

 

Does that make sense?