MAT01B1: Areas between curves (and some volumes) · MAT01B1: Areas between curves (and some...
Transcript of MAT01B1: Areas between curves (and some volumes) · MAT01B1: Areas between curves (and some...
![Page 1: MAT01B1: Areas between curves (and some volumes) · MAT01B1: Areas between curves (and some volumes) Dr Craig 5 September 2018](https://reader031.fdocuments.us/reader031/viewer/2022022108/5bf8a53509d3f294138c8478/html5/thumbnails/1.jpg)
MAT01B1:Areas between curves (and some volumes)
Dr Craig
5 September 2018
![Page 2: MAT01B1: Areas between curves (and some volumes) · MAT01B1: Areas between curves (and some volumes) Dr Craig 5 September 2018](https://reader031.fdocuments.us/reader031/viewer/2022022108/5bf8a53509d3f294138c8478/html5/thumbnails/2.jpg)
My details:
I Consulting hours:
Monday 14h40 – 15h25
Thursday 11h20 – 12h55
Friday 11h20 – 12h55
I Office C-Ring 508
https://andrewcraigmaths.wordpress.com/
(Or, just google ‘Andrew Craig maths’.)
![Page 3: MAT01B1: Areas between curves (and some volumes) · MAT01B1: Areas between curves (and some volumes) Dr Craig 5 September 2018](https://reader031.fdocuments.us/reader031/viewer/2022022108/5bf8a53509d3f294138c8478/html5/thumbnails/3.jpg)
Some new curves:
x = y2
x = y2 − 9
When sketching curves such as these, pay
attention to the sign (+ve or −ve) of x and
y and also the direction of the shifts. Plug in
x = 0 and y = 0 to get reference points.
![Page 4: MAT01B1: Areas between curves (and some volumes) · MAT01B1: Areas between curves (and some volumes) Dr Craig 5 September 2018](https://reader031.fdocuments.us/reader031/viewer/2022022108/5bf8a53509d3f294138c8478/html5/thumbnails/4.jpg)
The area between two curves
Height of rectangle centered at xi is given by:
(y-value of top curve) − (y-value bottom curve)
or, in this case:f (xi)− g(xi)
![Page 5: MAT01B1: Areas between curves (and some volumes) · MAT01B1: Areas between curves (and some volumes) Dr Craig 5 September 2018](https://reader031.fdocuments.us/reader031/viewer/2022022108/5bf8a53509d3f294138c8478/html5/thumbnails/5.jpg)
Example: find the area bounded above by
the curve y = ex, bounded below by y = x
and bounded on the left by x = 0 and on the
right by x = 1.
![Page 6: MAT01B1: Areas between curves (and some volumes) · MAT01B1: Areas between curves (and some volumes) Dr Craig 5 September 2018](https://reader031.fdocuments.us/reader031/viewer/2022022108/5bf8a53509d3f294138c8478/html5/thumbnails/6.jpg)
Example: find the area bounded by the
curves y = x2 and y = 2x− x2.
![Page 7: MAT01B1: Areas between curves (and some volumes) · MAT01B1: Areas between curves (and some volumes) Dr Craig 5 September 2018](https://reader031.fdocuments.us/reader031/viewer/2022022108/5bf8a53509d3f294138c8478/html5/thumbnails/7.jpg)
Area between curves
The area between two curves y = f (x)
and y = g(x), and between x = a and
x = b is
A =
∫ b
a
|f (x)− g(x)| dx
![Page 8: MAT01B1: Areas between curves (and some volumes) · MAT01B1: Areas between curves (and some volumes) Dr Craig 5 September 2018](https://reader031.fdocuments.us/reader031/viewer/2022022108/5bf8a53509d3f294138c8478/html5/thumbnails/8.jpg)
Example 5
Find the area of the region bounded by
y = sinx, y = cosx, x = 0 and x = π/2.
Solution: 2√2− 2.
![Page 9: MAT01B1: Areas between curves (and some volumes) · MAT01B1: Areas between curves (and some volumes) Dr Craig 5 September 2018](https://reader031.fdocuments.us/reader031/viewer/2022022108/5bf8a53509d3f294138c8478/html5/thumbnails/9.jpg)
Example 6
Find the area enclosed by the line y = x− 1
and the parabola y2 = 2x + 6.
Solution:∫ 4
−2
[(y + 1)−
(y2
2− 3
)]dy = 18
![Page 10: MAT01B1: Areas between curves (and some volumes) · MAT01B1: Areas between curves (and some volumes) Dr Craig 5 September 2018](https://reader031.fdocuments.us/reader031/viewer/2022022108/5bf8a53509d3f294138c8478/html5/thumbnails/10.jpg)
Using two methods: find the area between
the curves y2 = 4− x and 4y = −x + 4.
![Page 11: MAT01B1: Areas between curves (and some volumes) · MAT01B1: Areas between curves (and some volumes) Dr Craig 5 September 2018](https://reader031.fdocuments.us/reader031/viewer/2022022108/5bf8a53509d3f294138c8478/html5/thumbnails/11.jpg)
Integrating with respect to x we get:∫ 4
−12
(√4− x + x
4− 1)dx
Integrating with respect to y we get:∫ 4
0
(4y − y2) dy
In both cases the answer is32
3.
![Page 12: MAT01B1: Areas between curves (and some volumes) · MAT01B1: Areas between curves (and some volumes) Dr Craig 5 September 2018](https://reader031.fdocuments.us/reader031/viewer/2022022108/5bf8a53509d3f294138c8478/html5/thumbnails/12.jpg)
Using both methods: find the area of the
region between x + y2 = 4 and x− y = 2.
![Page 13: MAT01B1: Areas between curves (and some volumes) · MAT01B1: Areas between curves (and some volumes) Dr Craig 5 September 2018](https://reader031.fdocuments.us/reader031/viewer/2022022108/5bf8a53509d3f294138c8478/html5/thumbnails/13.jpg)
Solution: the problem on the previous slide
is much easier to solve if you integrate with
respect to y.
Either method will give you A = 412.
![Page 14: MAT01B1: Areas between curves (and some volumes) · MAT01B1: Areas between curves (and some volumes) Dr Craig 5 September 2018](https://reader031.fdocuments.us/reader031/viewer/2022022108/5bf8a53509d3f294138c8478/html5/thumbnails/14.jpg)
Volumes
![Page 15: MAT01B1: Areas between curves (and some volumes) · MAT01B1: Areas between curves (and some volumes) Dr Craig 5 September 2018](https://reader031.fdocuments.us/reader031/viewer/2022022108/5bf8a53509d3f294138c8478/html5/thumbnails/15.jpg)
Familiar volume calculations
We can calculate the volume of many shapes
by multiplying the area of the base by the
height of the shape. However, this only works
if the shape at the base is extended upwards
at right angles and remains constant.
![Page 16: MAT01B1: Areas between curves (and some volumes) · MAT01B1: Areas between curves (and some volumes) Dr Craig 5 September 2018](https://reader031.fdocuments.us/reader031/viewer/2022022108/5bf8a53509d3f294138c8478/html5/thumbnails/16.jpg)
When calculating the area of a disk at x, we
will use the notation A(x).
We will sometimes integrate with respect to
y and then we will write A(y) for the area of
a disk at y.
![Page 17: MAT01B1: Areas between curves (and some volumes) · MAT01B1: Areas between curves (and some volumes) Dr Craig 5 September 2018](https://reader031.fdocuments.us/reader031/viewer/2022022108/5bf8a53509d3f294138c8478/html5/thumbnails/17.jpg)
Calculating the volume of a sphere:
![Page 18: MAT01B1: Areas between curves (and some volumes) · MAT01B1: Areas between curves (and some volumes) Dr Craig 5 September 2018](https://reader031.fdocuments.us/reader031/viewer/2022022108/5bf8a53509d3f294138c8478/html5/thumbnails/18.jpg)
Calculating the volume of a sphere: