Areas between curves

1. f
g
a b
( ) ( ) ( ) ( )
b b b
a a a
f x dx g x dx f x g x dx
  
  
Areas Between Curves

2. f
g
a b
( ) ( )
b
a
f x dx g x dx


What if the green graph falls below the x-axis somewhere?
Do the arrows match the equation we came up with before?

3. f
g
a
b
( ) ( )
b
a
f x g x dx


What if both curves lie partially below the x-axis?

4. f
g
a b
( ) ( )
b
a
f x g x dx


What if both curves lie entirely below the x-axis?

5. In Summary
If the graph of the function f lies above the graph of g
on the interval [a,b], the integral
gives us actual geometric area between the graphs of g
and f, whether or not one or both of the functions
drops below the x-axis.
( ) ( )
b
a
f x g x dx



6. So what do we do?
• We find the points of intersection of the
graphs of f and g. That is, we solve the
equation f (x)=g(x) for x.
• We determine which of the two functions is
above and which is below. (How might we
do this?)
• Then we compute the integral.

7. True or false
We don’t really have to determine which function is
above and which is below. We can just compute
and take the absolute value of the result. (Since the
answer will obviously just be “off” by a minus
sign if we subtract the wrong function.)
( ) ( )
b
a
f x g x dx



8. Well, sort of, but you have to be
Cautious. . .
f
g
a b c
f
g
a b
Does not work here!
Works here!