area between curvescdsapcalcab2014.weebly.com/.../area_between_curves.pdfarea between curves (2/3)...
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![Page 1: Area Between Curvescdsapcalcab2014.weebly.com/.../area_between_curves.pdfArea Between Curves (2/3) Applying the difference of two functions, , as the height of each rectangle, we obtain](https://reader035.vdocument.in/reader035/viewer/2022071106/5fe057bfc9b62f6a4e453e40/html5/thumbnails/1.jpg)
Area Between Curves
Lesson #2 of
Unit 6. Applications of Integration
(Textbook 6.1)
![Page 2: Area Between Curvescdsapcalcab2014.weebly.com/.../area_between_curves.pdfArea Between Curves (2/3) Applying the difference of two functions, , as the height of each rectangle, we obtain](https://reader035.vdocument.in/reader035/viewer/2022071106/5fe057bfc9b62f6a4e453e40/html5/thumbnails/2.jpg)
Area Between Curves (1/3)
Previously, we have considered an area under a curve using Riemann Sum, and obtained the result…
Now we consider a region between curves.
Each rectangle’s height of Riemann Sum for an area under a curve was the value of .
Then for an area between two curves, the height is the difference between two curves, which is as shown on the left.
Text p.422
![Page 3: Area Between Curvescdsapcalcab2014.weebly.com/.../area_between_curves.pdfArea Between Curves (2/3) Applying the difference of two functions, , as the height of each rectangle, we obtain](https://reader035.vdocument.in/reader035/viewer/2022071106/5fe057bfc9b62f6a4e453e40/html5/thumbnails/3.jpg)
Area Between Curves (2/3)
Applying the difference of two functions, , as the height of each rectangle, we obtain following result for the calculation of area between two curves.
Above is the definite integral of f – g, so we generalize it as following.
Text p.422
![Page 4: Area Between Curvescdsapcalcab2014.weebly.com/.../area_between_curves.pdfArea Between Curves (2/3) Applying the difference of two functions, , as the height of each rectangle, we obtain](https://reader035.vdocument.in/reader035/viewer/2022071106/5fe057bfc9b62f6a4e453e40/html5/thumbnails/4.jpg)
Example
Find the area of the region bounded above by y = e x, bounded below by y = x, and bounded on the sides by x = 0 and x = 1.
Sketch the region enclosed by the parabolas and , and then find the area.
Text p.423
![Page 5: Area Between Curvescdsapcalcab2014.weebly.com/.../area_between_curves.pdfArea Between Curves (2/3) Applying the difference of two functions, , as the height of each rectangle, we obtain](https://reader035.vdocument.in/reader035/viewer/2022071106/5fe057bfc9b62f6a4e453e40/html5/thumbnails/5.jpg)
Practice
Sketch the region enclosed by the curves, and then find the area.
Text p.600
![Page 6: Area Between Curvescdsapcalcab2014.weebly.com/.../area_between_curves.pdfArea Between Curves (2/3) Applying the difference of two functions, , as the height of each rectangle, we obtain](https://reader035.vdocument.in/reader035/viewer/2022071106/5fe057bfc9b62f6a4e453e40/html5/thumbnails/6.jpg)
Area Between Curves (3/3)
Some regions are best treated by regarding x as a function of y. If a region is bounded by curves with equations x = f (y), x = g (y), y = c, and y = d, where f and g are continuous and f (y) g (y) for c y d, then its area is…
Text p.426
Find the area of the region enclosed by the parabolas and
![Page 7: Area Between Curvescdsapcalcab2014.weebly.com/.../area_between_curves.pdfArea Between Curves (2/3) Applying the difference of two functions, , as the height of each rectangle, we obtain](https://reader035.vdocument.in/reader035/viewer/2022071106/5fe057bfc9b62f6a4e453e40/html5/thumbnails/7.jpg)
Practice
Sketch the region enclosed by the curves, and then find the area.
Text p.427
![Page 8: Area Between Curvescdsapcalcab2014.weebly.com/.../area_between_curves.pdfArea Between Curves (2/3) Applying the difference of two functions, , as the height of each rectangle, we obtain](https://reader035.vdocument.in/reader035/viewer/2022071106/5fe057bfc9b62f6a4e453e40/html5/thumbnails/8.jpg)
Practice
Use calculus to find the area of the triangle with the given vertices.
(0, 0), (3, 1), (1, 2)
Text p.427
![Page 9: Area Between Curvescdsapcalcab2014.weebly.com/.../area_between_curves.pdfArea Between Curves (2/3) Applying the difference of two functions, , as the height of each rectangle, we obtain](https://reader035.vdocument.in/reader035/viewer/2022071106/5fe057bfc9b62f6a4e453e40/html5/thumbnails/9.jpg)
Area Between Curves: The Split Case
If we are asked to find the area between the curves y = f (x) and y = g (x) where f (x) g (x) for some values x of but g (x) f (x) for other values of x, then we split the given region S into several regions S1, S2, …
We then define the area of the region S to be the sum of the areas of the smaller regions S1, S2, ... Since…
f (x) – g (x) when f (x) g (x)
| f (x) – g (x) | =
g (x) – f (x) when g (x) f (x)
the area between the curves y = f (x) and y = g (x) and between x = a and x = b is…
Text p.426
![Page 10: Area Between Curvescdsapcalcab2014.weebly.com/.../area_between_curves.pdfArea Between Curves (2/3) Applying the difference of two functions, , as the height of each rectangle, we obtain](https://reader035.vdocument.in/reader035/viewer/2022071106/5fe057bfc9b62f6a4e453e40/html5/thumbnails/10.jpg)
Example
Find the area of the region bounded by the curves , , , and .
Text p.423
![Page 11: Area Between Curvescdsapcalcab2014.weebly.com/.../area_between_curves.pdfArea Between Curves (2/3) Applying the difference of two functions, , as the height of each rectangle, we obtain](https://reader035.vdocument.in/reader035/viewer/2022071106/5fe057bfc9b62f6a4e453e40/html5/thumbnails/11.jpg)
Practice
Sketch the region enclosed by the curves, and then find the area.
Text p.600
![Page 12: Area Between Curvescdsapcalcab2014.weebly.com/.../area_between_curves.pdfArea Between Curves (2/3) Applying the difference of two functions, , as the height of each rectangle, we obtain](https://reader035.vdocument.in/reader035/viewer/2022071106/5fe057bfc9b62f6a4e453e40/html5/thumbnails/12.jpg)
Suggested Problems
Textbook p.427
Sketch the region and find the area: 1, 5, 8, 16
Sketch the region and find the area: 2, 12, 20, 30
Sketch the region and find the area: 21, 22, 27
![Page 13: Area Between Curvescdsapcalcab2014.weebly.com/.../area_between_curves.pdfArea Between Curves (2/3) Applying the difference of two functions, , as the height of each rectangle, we obtain](https://reader035.vdocument.in/reader035/viewer/2022071106/5fe057bfc9b62f6a4e453e40/html5/thumbnails/13.jpg)
Suggested Problems - SOLUTIONS -
Textbook p.427
1) 5)
8) 16)
2) 12)
20) 30)
![Page 14: Area Between Curvescdsapcalcab2014.weebly.com/.../area_between_curves.pdfArea Between Curves (2/3) Applying the difference of two functions, , as the height of each rectangle, we obtain](https://reader035.vdocument.in/reader035/viewer/2022071106/5fe057bfc9b62f6a4e453e40/html5/thumbnails/14.jpg)
Suggested Problems - SOLUTIONS -
Textbook p.427
21) OR
by symmetry
22) By symmetry,
27)