Notes to BC students:• I hope everyone had great holidays, I did,
including experiencing a blizzard, but now I’m sick…
• Since we missed the time before the holidays, some Unit 6 topic(s) will be moved to Quarter III.
• This applies to both morning and afternoon classes.
• The problem is to find the area between two curves, so we start with a couple of friendly calculus curves.
The first is , or . y =x2 f (x)
Since h is the distance from the upper to lower curve. This is simply the difference of the two y-coordinates.
h =yupper−ylowerorh=f(x) − g(x)
This means that h(x) = x −x2 .
• We can find the total area between the curves by integrating h between the points of intersection.
We can evaluate the integral using the Fundamental Theorem of the Calculus.
x −x2
( )dx0
1
∫ =
=
23x
32 −
13x3⎛
⎝⎜⎞⎠⎟
0
1
=23
−13
− 0 − 0( ) =13
.
As a second example, find the area between
f (y) =x= y3 and g(y)=x =2y2 .
First, we need to graph the functions and see the defined area.
Later, we will find the intersection. First, we define h.
h =xright −xleft =f(y) − g(y) = y3 − 2y2
h
f
g
Notice that h is the difference between the two x-coordinates.
Notice this distance uses coordinates from the right function minus coordinates from the left function.
To have distance be a positive number one must always subtract a smaller from a larger one.
As with the first example we integrate h from beginning to end. We see that the origin is one point of intersection.
We need to find the other point of intersection.
y3 =2y2 (cubing each side)
y=8y6
y5 =18
y= 18
5 ≈0.65975
Finally, the area is
h(y)dy
0
18
5
∫ = y3 −2y2( )dy0
18
5
∫ ≈0.239
This is a good time to use your calculator!
Note that in this example the limits of integration are y-values, and the integrand is a function of y.
There are several points that should be made:
• Graph the functions.
• Decide whether you will work in vertical or horizontal distances. Use the one that it easiest for the problem. n.b. This is not always x!
• Distance is always positive, remember to subtract the smaller value from the larger one, whether using x or y.