area region of a graph

Question

Find the area of the region bounded by the graphs of f(x)=5-x^2 and g(x)=x^2 -3

Answer 1

area between the curves = 21 1/3 square units

x^2-3 = 5-x^2 set them equal to find limits of integration

x=+/-2

integrate f(x) - g(x) dx

= integral (5-x^2 - (x^2 -3))= -2x^2+8

=2 times integral of 4-x^2 evaluated from -2 to 2

= 2(4x -x^3/3) from -2 to 2

= 2(8- 8/3 +8 -8/8)

= 2(16-16/3)

= 32-32/3=96/3-32/3

=64/3

=21 1/3 un^2=area

Answer 2

Hi again Ana A.,

First, set f(x) and g(x) equal to each other and solve for the interval over which f(x) is greater than g(x). Why do I seek that, rather than the other choice? It is because at extreme values of x (either positive or negative), -x^2 will be negative vs. +x^2. Therefore, if there is any region that is bounded by f(x) - g(x), it will be in the middle somewhere. (And there is such a region!)

Why don't you sketch the two curves, using a few points each, to convince yourself of that.

So now you know the endpoints of your integration region. Set up the integral of the difference -- it is the difference of the integrals of each over that same interval, but you can certainly combine the algebraic formulas and just integrate that expression, if you like.

-- Cheers, --Mr. d.