Download Calc Chapter 7 notes

Chapter 7
AP Calculus BC
7.1 Integral as Net Change
ds  v(t )  t 2  8
dt
(t 1)2
Models a particle moving along the xaxis for t from 0 to 5.
What is its initial velocity? When does it stop? v(5) = ???
Given s(0) = 9, find its position at t = 1 and t =5….
Total Distance traveled:
you need……
s(0) 
s(anytime v = 0)=
s(final time) 
This problem you need s(0),
s(1.25), and s(5)
On calculator, use absolute value.
Hooke’s Law (springs) – F = kx
k = force constant for that spring
Example 7 p. 385 x = distance you want to stretch or compress
Work = F d
7.2 Areas in the Plane
Area between the curves – If f and g are cts. Functions with f(x) >g(x)
throughout the interval [a,b] then the area between the curves is:
b
A   ( f ( x)  g ( x))dx
a
Examples:
Top - Bottom
y  sec2 x
y  sin x
y  2  x2
y  x
y  cos x
y  x2 1
Integrating with respect to y (Right – Left)
Choosing:
(Both ways)
y  x3
x  y2  2
y x
y  x2
bounded by x-axis
y x
y  x2
7.3 Volumes (Cross Sections)
Slicing a Cross – Section – Method:
1.
2.
3.
4.
Sketch the solid and a typical cross section (A(x))
Find a formula for A(x)
Find the Limits of Integration
Integrate A(x) to find the Volume.
Examples: Walk through #2 p. 406
Area
Formulas:
A  L W
A  s2
A  3 s2
4
7.3 Volumes - Disks/Washers
Two Methods for rotating around an axis, thus creating a volume.
Question: Is the shaded region flat against the rotation axis?
YES – Disk Method   r 2dx
NO – Washer Method   (R2  r 2)dx
Examples: rotate around x-axis
y  ( x  3)2
y  cos x
y  sin x
1st bounded area
If you switch the
rotation axis to
something other than
the axes (x or y). That
number must included
in all Radii.
7.3 Cylindrical Shell Method
In the shell method:
If you rotate around the x-axis – you use y-values.
If you rotate around the y-axis – you use x-values.
Formulas :
y x
x4
about the x-axis
2  (radius)(Height)dx
2  (radius)(Height)dy
Around y – axis
Around x - axis
If you change the rotational axis,
ONLY the radius changes!!!!!!!!!!
Problem # 33 in book………
7.4 Lengths of Curves
Using a Riemann sum derive the length of curve formula.
p.412-13…….
Arc Length – Length of a Smooth Curve:
If a smooth curve begins at (a,c) and ends at (b,d), then
the length of the curve is :
b
L   1 ( dy )2 dx
a
dx
Examples:
3
4
2
y
x 2 1
3
on[0,1]
y  cos x
on[0,  ]
4
or
d
L   1 ( dx )2 dy
c
dy
7.5 Science Applications
Work is Force(in the direction of motion) times displacement. W  Fd
Hooke’s Law: Force to stretch or compress a spring x – units,
from its natural length is a constant times x.
F  Kx
It takes a force of 10N to stretch a spring 2 m beyond its
natural length. How much work is done in stretching the
spring 6 m from its natural length?
F  Kx
 Fdx
A leaky bucket weighs 22 N empty. It is lifted from the ground at
a constant rate to a pt. 20 m above the ground by a rope
weighing .4N/m. The bucket starts with 70 N of water but it
leaks at a constant rate and just finishes draining as the bucket
reaches the top. Find the Amount of Work done.
Leaky Bucket Solution:
Lifting the bucket alone:
22 N 20m
20
Lifting the water alone:
0
Lifting the rope alone:
20
All together:
0
70( 20  x )dx
20
0.4(20  x)dx
440 + 700 + 80 = 1220 Nm
440 Nm
700 Nm
80 Nm