AP Calculus AB
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Transcript of AP Calculus AB
![Page 1: AP Calculus AB](https://reader036.fdocuments.in/reader036/viewer/2022062718/56812c71550346895d910a1d/html5/thumbnails/1.jpg)
04/19/23 Perkins
AP Calculus AB
Day 5Section 4.2
![Page 2: AP Calculus AB](https://reader036.fdocuments.in/reader036/viewer/2022062718/56812c71550346895d910a1d/html5/thumbnails/2.jpg)
x
1
left endptn
i
f x
a
Area Under a CurveFind the area of the region bounded by y = f(x), the x-axis, x = a, and x = b.
b
4
2
5a b
4
2
5
We call it the Lower Sum. We call it the Upper Sum.
Approximate the area by creating rectangles of equal width whose
endpoints are on f(x).
x
Each right endpoint is on f(x)Each left endpoint is on f(x)
This over-estimates the area under the curve…
This under-estimates the area under the curve…
1
n
i
f a i x x
1
right endptn
i
f x
1
1n
i
f a i x x
n = # of rectangles
1
height widthn
i
A
Each method is called a Riemann Sum.
![Page 3: AP Calculus AB](https://reader036.fdocuments.in/reader036/viewer/2022062718/56812c71550346895d910a1d/html5/thumbnails/3.jpg)
lower sum actual area upper sum
lim lower sum actual area lim upper sumn n
1
limn
ni
f a i x x
How do we make these approximations for the area under a curve more accurate?
1
lim right endptn
ni
f x
Use more rectangles.
(Always choose whichever sum involves right endpoints.)
The Limit Definition for finding the area under a curve:
or
where b a
xn
![Page 4: AP Calculus AB](https://reader036.fdocuments.in/reader036/viewer/2022062718/56812c71550346895d910a1d/html5/thumbnails/4.jpg)
Find the area beneath (above the x-axis) in the interval [1,3].
heightA x
2 3y x
a. Use 1 rectangle.8
6
4
2
2 4
3 2f 9 218
b. Use 2 rectangles.
8
6
4
2
2 4
3 12
1x
3 11
2x
2 3A f x f x
7 1 9 1 16
If a specific number of rectangles is given, it is often easier to find the area without using sigma!
![Page 5: AP Calculus AB](https://reader036.fdocuments.in/reader036/viewer/2022062718/56812c71550346895d910a1d/html5/thumbnails/5.jpg)
Find the area beneath (above the x-axis) in the interval [1,3].
2 3y x
c. Use the limit definition.8
6
4
2
2 4
1
limn
ni
A f a i x x
3 1x
n
n
right endpt a i x 2
1 in
2
n
21
i
n
2right endpt 2 1 3inf
45
i
n
1
4 2lim 5
n
ni
i
n n
1 3
![Page 6: AP Calculus AB](https://reader036.fdocuments.in/reader036/viewer/2022062718/56812c71550346895d910a1d/html5/thumbnails/6.jpg)
1
4 2lim 5
n
ni
iA
n n
1
2 4lim 5
n
ni
i
n n
1 1
2 4 2lim 5
n n
ni i
i
n n n
21 1
8 2lim 5
n n
ni i
in n
2
18 2lim 5
2n
n nn
n n
2
2
8lim 10
2n
n n
n
2
2
8 8lim lim 10
2n n
n n
n
4 10
14
![Page 7: AP Calculus AB](https://reader036.fdocuments.in/reader036/viewer/2022062718/56812c71550346895d910a1d/html5/thumbnails/7.jpg)
Perkins
AP Calculus AB
Day 5Section 4.2
![Page 8: AP Calculus AB](https://reader036.fdocuments.in/reader036/viewer/2022062718/56812c71550346895d910a1d/html5/thumbnails/8.jpg)
a
Area Under a CurveFind the area of the region bounded by y = f(x), the x-axis, x = a, and x = b.
b
4
2
5a b
4
2
5
Approximate the area by creating rectangles of equal width whose
endpoints are on f(x).
![Page 9: AP Calculus AB](https://reader036.fdocuments.in/reader036/viewer/2022062718/56812c71550346895d910a1d/html5/thumbnails/9.jpg)
How do we make these approximations for the area under a curve more accurate?
The Limit Definition for finding the area under a curve:
![Page 10: AP Calculus AB](https://reader036.fdocuments.in/reader036/viewer/2022062718/56812c71550346895d910a1d/html5/thumbnails/10.jpg)
Find the area beneath (above the x-axis) in the interval [1,3].
2 3y x
a. Use 1 rectangle.8
6
4
2
2 4
b. Use 2 rectangles.
8
6
4
2
2 4
![Page 11: AP Calculus AB](https://reader036.fdocuments.in/reader036/viewer/2022062718/56812c71550346895d910a1d/html5/thumbnails/11.jpg)
Find the area beneath (above the x-axis) in the interval [1,3].
2 3y x
c. Use the limit definition.8
6
4
2
2 4
![Page 12: AP Calculus AB](https://reader036.fdocuments.in/reader036/viewer/2022062718/56812c71550346895d910a1d/html5/thumbnails/12.jpg)