finding the area between curves application of integration
TRANSCRIPT
![Page 1: Finding the Area Between Curves Application of Integration](https://reader033.vdocuments.mx/reader033/viewer/2022051401/56649ea25503460f94ba5d06/html5/thumbnails/1.jpg)
Finding the Area Between Curves
Application of Integration
![Page 2: Finding the Area Between Curves Application of Integration](https://reader033.vdocuments.mx/reader033/viewer/2022051401/56649ea25503460f94ba5d06/html5/thumbnails/2.jpg)
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.
![Page 3: Finding the Area Between Curves Application of Integration](https://reader033.vdocuments.mx/reader033/viewer/2022051401/56649ea25503460f94ba5d06/html5/thumbnails/3.jpg)
• 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)
![Page 4: Finding the Area Between Curves Application of Integration](https://reader033.vdocuments.mx/reader033/viewer/2022051401/56649ea25503460f94ba5d06/html5/thumbnails/4.jpg)
• And the second is g(x), or y = x.
![Page 5: Finding the Area Between Curves Application of Integration](https://reader033.vdocuments.mx/reader033/viewer/2022051401/56649ea25503460f94ba5d06/html5/thumbnails/5.jpg)
• A closer look:
![Page 6: Finding the Area Between Curves Application of Integration](https://reader033.vdocuments.mx/reader033/viewer/2022051401/56649ea25503460f94ba5d06/html5/thumbnails/6.jpg)
• We are interested in finding the area of the purple region.
![Page 7: Finding the Area Between Curves Application of Integration](https://reader033.vdocuments.mx/reader033/viewer/2022051401/56649ea25503460f94ba5d06/html5/thumbnails/7.jpg)
• Let h be the distance between the two curves.
h
![Page 8: Finding the Area Between Curves Application of Integration](https://reader033.vdocuments.mx/reader033/viewer/2022051401/56649ea25503460f94ba5d06/html5/thumbnails/8.jpg)
• Notice how h changes as we move from left to right.
h
![Page 9: Finding the Area Between Curves Application of Integration](https://reader033.vdocuments.mx/reader033/viewer/2022051401/56649ea25503460f94ba5d06/html5/thumbnails/9.jpg)
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 .
![Page 10: Finding the Area Between Curves Application of Integration](https://reader033.vdocuments.mx/reader033/viewer/2022051401/56649ea25503460f94ba5d06/html5/thumbnails/10.jpg)
• We can find the total area between the curves by integrating h between the points of intersection.
![Page 11: Finding the Area Between Curves Application of Integration](https://reader033.vdocuments.mx/reader033/viewer/2022051401/56649ea25503460f94ba5d06/html5/thumbnails/11.jpg)
• Note that the two curves intersect at the origin and at (1,1).
![Page 12: Finding the Area Between Curves Application of Integration](https://reader033.vdocuments.mx/reader033/viewer/2022051401/56649ea25503460f94ba5d06/html5/thumbnails/12.jpg)
The area between the curves is
A = h(x)dx
0
1
∫The 0 and 1 are the starting and ending values of x.
![Page 13: Finding the Area Between Curves Application of Integration](https://reader033.vdocuments.mx/reader033/viewer/2022051401/56649ea25503460f94ba5d06/html5/thumbnails/13.jpg)
Further,
The area is
x −x2( )
0
1
∫ dx.
![Page 14: Finding the Area Between Curves Application of Integration](https://reader033.vdocuments.mx/reader033/viewer/2022051401/56649ea25503460f94ba5d06/html5/thumbnails/14.jpg)
We can evaluate the integral using the Fundamental Theorem of the Calculus.
x −x2
( )dx0
1
∫ =
=2
3−
1
3− 0 − 0( ) =
1
3.
![Page 15: Finding the Area Between Curves Application of Integration](https://reader033.vdocuments.mx/reader033/viewer/2022051401/56649ea25503460f94ba5d06/html5/thumbnails/15.jpg)
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.
![Page 16: Finding the Area Between Curves Application of Integration](https://reader033.vdocuments.mx/reader033/viewer/2022051401/56649ea25503460f94ba5d06/html5/thumbnails/16.jpg)
x = y3 and x = 2y2
f
g
![Page 17: Finding the Area Between Curves Application of Integration](https://reader033.vdocuments.mx/reader033/viewer/2022051401/56649ea25503460f94ba5d06/html5/thumbnails/17.jpg)
Zooming in:
Notice that the upper intersection is not made of simple values.
fg
![Page 18: Finding the Area Between Curves Application of Integration](https://reader033.vdocuments.mx/reader033/viewer/2022051401/56649ea25503460f94ba5d06/html5/thumbnails/18.jpg)
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.
![Page 19: Finding the Area Between Curves Application of Integration](https://reader033.vdocuments.mx/reader033/viewer/2022051401/56649ea25503460f94ba5d06/html5/thumbnails/19.jpg)
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.
![Page 20: Finding the Area Between Curves Application of Integration](https://reader033.vdocuments.mx/reader033/viewer/2022051401/56649ea25503460f94ba5d06/html5/thumbnails/20.jpg)
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
![Page 21: Finding the Area Between Curves Application of Integration](https://reader033.vdocuments.mx/reader033/viewer/2022051401/56649ea25503460f94ba5d06/html5/thumbnails/21.jpg)
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.
![Page 22: Finding the Area Between Curves Application of Integration](https://reader033.vdocuments.mx/reader033/viewer/2022051401/56649ea25503460f94ba5d06/html5/thumbnails/22.jpg)
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.