34

Calculus 2 Name:____________________________________

Lesson- Homogeneous Differential Equations

Date:_____________________________________

Objective: To verify Homogeneity and to solve Homogeneous Differential Equations.

A homogeneous differential equation is an equation of the form: M(x,y)dx + N(x,y)dy = 0 where M and N are

homogeneous functions of the same degree.

How to determine if functions are homogeneous (and determine the degree):

Verify

€

f (tx,ty) = tnf (x,y) and state n.

What this means:

Take the original function and replace the x & y with tx & ty.

Simplify and factor out a GCF.

What remains should be

€

tn⋅ the expression that is the original implicitly defined function.

The n is the degree.

How to solve a homogeneous differential equation:

If M(x,y)dx + N(x,y)dy = 0 is Homogeneous, then it can be transformed into a differential equation whose

variables are separable by the substitution: y = vx where v is a differentiable function of x.

Sub out an aspect of y using y = vx

Note: you will need to find y’

Simplify

Separate variables and integrate

Sub out any aspect of v using y=vx

Examples:

1. Find the general solution of

€

(x2− y

2)dx + 3xydy = 0

35

2. Solve the homogeneous differential equation:

€

y'=x3

+ y3

xy2

3. Solve the homogeneous differential equation:

€

y'=2x + 3y

x

37

36

Calculus 2 Name:____________________________________

Lesson- Area Between Curves

Date:_____________________________________

How to find the area between two curves on [a, b]:

• Find the point(s) of intersection.

• Establish which function is dominant from a to the point of intersection.

• Integrate the difference between the dominant and non-dominant functions, respectively.

• Establish which function is dominant from the point of intersection to b.

• Integrate the difference between the dominant and non-dominant functions, respectively.

• Add these two positive areas together to get the total area.

• If looking for average: Divide your answer by b – a.

1. Find the area of the region bounded between and .

38

37

2. Find the area of the region bounded by , the x and y axes, and

a. With respect to x. b. With respect to y.

3. Find the area of the region in quadrant I bounded by , , and

a. With respect to x. b. With respect to y.

39

38

Calculus 2 Name:____________________________________

Lesson- Volumes of Cross Sections

Date:_____________________________________

Objective: To find the volume of a solid given area boundaries of the base and known cross-sections

DO NOW: Find the area of the region bounded by: , the axes, and the line x = 3.

---------------------------------------------------------------------------------------------------------------------------------------

What if…

…emerging from the area you calculated were infinitely many squares?

What would be the volume of this new solid?

---------------------------------------------------------------------------------------------------------------------------------------

Now try these…

1. The base of solid S is the region enclosed by the graph of , the line x = e, and the x-axis. If

the cross-sections of S perpendicular to the x-axis are semi-circles, then the volume of S is…?

40

39

2. The base of solid S is the region enclosed by the graphs , , and x = 0. If

the cross-sections perpendicular to the x-axis are equilateral triangles, find the volume of the solid.

3. Let R be the region enclosed by graphs of and y = cos x.

a. Find the area of R (use your GC)

b. The base of a solid is region R. Each cross section of the solid perpendicular to the x-axis is an

equilateral triangle. Write an expression involving one or more integrals that gives the volume

of the solid. Do not evaluate.

41

42

Calculus 2 Name:____________________________________ Lesson- Volumes of Discs Date:_____________________________________ Objective: To find the volume of a solid using the disc method. DO NOW: Write an expression to find the area of the region bounded by the axes, , and x = 8 in Q1. --------------------------------------------------------------------------------------------------------------------------------------- What if… …this area was rotated around the x-axis? What would be the volume of this new figure?

…this area was rotated around the line x = 8? What would be the volume of this new figure?

3 xy

43

Now try these…

1. The region in the first quadrant bounded by the graphs of y = sec x , , and the axes is rotated

about the x-axis. What is the volume of the solid generated? 2. The volume of a solid obtained by revolving the region enclosed by the ellipse about the x- axis is? 3. Let R be the region in the first quadrant bounded by the graphs of , the line y = 6, and the y- axis. a. Find the area of R. b. Write, but do not evaluate, an integral expression that gives the volume of the solid generated when R is rotated about the line y = 6.

4S

x

99 22 � yx

xy 2

44

Calculus 2 Name:____________________________________ Lesson- Volumes of Washers Date:_____________________________________ Objective: To find the volume of a solid using the washer method. Let’s get right to it… What if you have a gap between the given area and the axis of revolution???

Let R be the region between the graphs of y = 1 and y = sin x from x = 0 to x = . What is the volume of the

solid obtained by revolving R about the x-axis? --------------------------------------------------------------------------------------------------------------------------------------- Now try these… 1. The region enclosed by the graph of , the line x = 2, and the x-axis is revolved about the y-axis. What is the volume of the solid generated?

2S

2xy

45

2. A region in the first quadrant is enclosed by the graphs of , x = 1, and the coordinate axes. If the region is rotated about the y-axis, what is the volume of the solid generated? 3. Let R be the region in the first quadrant is enclosed by the graphs of , x = 7, and the coordinate axes. If the region is rotated about the y-axis, what is the volume of the solid generated?

xey 2

� � 3/11� xy

44

Calculus 2 Name:____________________________________

Lesson- Volumes of Cylindrical Shells

Date:_____________________________________

Objective: To find the volume of a solid using the shell method.

Let’s get right to it…

Sometimes using the washer method can be a pain. This happens when you have to develop 2 or more integrals

to describe the bounded volume; instead of doing that, we can use the Cylindrical Shell Method.

To do this we must create a rectangle PARALLEL to the axis of revolution and then find the volume of the

CYLINDER that has that rectangle as its HEIGHT. The distance that the rectangle is away from the axis of

revolution is the cylinder’s RADIUS. So knowing those pieces of info and the fact that the volume of a

cylinder is , we can now find the volume of a cylindrical “shell”

Examples

1. If p is defined as the distance from the axis of revolution and the center of the rectangular strip

then what is the radius of the outer cylinder?

2. What is the volume of the outer cylinder?

3. What is the radius and volume of the inner cylinder?

4. What is the volume of the Cylindrical Shell?

46

45

The Shell Method:

Vertical Axis of Revolution Horizontal Axis of Revolution

where p(x) or p(y) represent the radius of the cylindrical shell and h(x) or h(y) represent the height of the strip.

47

48

So let’s revisit some examples in the Washer Lesson and, this time, apply the Cylindrical Shell Method. 1. The region enclosed by the graph of , the line x = 2, and the x-axis is revolved about the y-axis. What is the volume of the solid generated? 2. A region in the first quadrant is enclosed by the graphs of , x = 1, and the coordinate axes. If the region is rotated about the y-axis, what is the volume of the solid generated? 3. Let R be the region in the first quadrant is enclosed by the graphs of , x = 7, and the coordinate axes. If the region is rotated about the y-axis, what is the volume of the solid generated?

2xy

xey 2

� � 3/11� xy

49

Calculus 2 Name:____________________________________ Practice- Volumes of Revolution Date:_____________________________________ Volume Problems- Definitely need to go to separate paper here! The goal is to find the VOLUME of a shape by adding up the volumes of slices. In general, the formula is

.

Examples (Calculator Intensive) 1. Find the volume of a solid created by revolving the line y = 4x on [2, 4] over the x-axis. 2. Find the volume of a solid created by revolving the lines y = e-x and y = x on [2, 4] over the x-axis. 3. Find the volume of the solid whose base is the region in quadrant I that is bounded by , , and

. All cross-sections perpendicular to the x-axis are a. Rectangles with height twice the width. b. Semi-circles. 4. The volume of the solid generated by rotating the region bounded by , , between and

around the x-axis 5. The volume of the solid generated by rotating the region in quadrant I bounded by and

around the x-axis. 6. The volume of the solid generated by rotating the region bounded by , , and around the y-axis. 7. The volume of the solid generated by rotating the region in quadrant I bounded by, ,x = 1 and

around a. The line . b. The line . c. The line .

³b

adxxAREA )(

3y x 0y 2x

3xy 0y 0x 4x

siny x cosy x

4y x � 0x 3y

2 1y x �10y

10y 3x 1y �

Volume Problems

50

Calculus 2 Name:____________________________________ Lesson- Applications of Integration: Arc Length, Surface Area Date:_____________________________________ Objective: To learn how to determine arc length of any function on [a, b] To learn how to find the area of the surface of a solid of revolution Definition of Arc Length: Let y = f(x) represent a smooth curve on the interval [a, b]. The arc length between a and b is:

Also for a curve given by x = g(y) on [c, d]:

Examples 1. Find the length of the arc connecting on f(x) = mx + b

2. Find the length of the arc of the graph of on the interval [0.5, 2]

3. Find the length of the arc of the graph of on the interval

�

s 1� [ f '(x)]2 dxa

b

³

�

s 1� [g'(y)]2 dyc

d

³

�

x1,y1� � & x2,y2� �

�

y x3

6�

12x

�

y ln(cosx)

�

0,S4

ª�¬�«�

º�¼�»�

51

Definition of the Area of a Surface of Revolution: Let y = f(x) have a continuous derivative on [a, b]. The area S of the surface of revolution formed by revolving he graph of f around a horizontal or vertical axis is:

where r(x) is the distance between the graph of f and the axis of revolution.

Also for a curve given by x = g(y) on [c, d]:

where r(x) is the distance between the graph of f and the axis of revolution.

Examples: 1. Find the surface area formed by revolving the graph of on the interval [0,1] about the x-axis. 2. Find the surface area formed by revolving the graph of

�

f (x) x 2 on [0,

�

2 ] about the y-axis.

�

S 2S r(x) 1� [ f '(x)]2 dxa

b

³�

S 2S r(y) 1� [g'(y)]2 dyc

d

³

�

f (x) x 3

y

52

Calculus 2 Name:____________________________________ Lesson- Applications of Integration: Fluid Pressure and Fluid Force Date:_____________________________________ Objective: To learn how to use integration to determine fluid pressure and fluid force. Definition of Fluid Pressure The pressure on an object at depth h in a liquid is: Pressure = P = wh Where w is the weight-density of the liquid per unit of volume Here are some common weight-densities of fluids in pounds per square foot. Ethyl Alcohol 49.4 Gasoline 41.0 – 43.0 Glycerin 78.6 Kerosene 51.2 Mercury 849.0 Seawater 64.0 Water 62.4 Pascal’s Principle states that the pressure exerted by a fluid at depth h is transmitted equally in all directions. Because fluid pressure is given in force per unit area (P = F/A), the fluid force on an object at a given depth would be the pressure times the area of the object:

�

F PA There are two variations of this idea

1) Horizontal orientation (downward force equivalent throughout) 2) Vertical orientation (downward force varies depending on location)

Good video explanation: http://www.sophia.org/fluid-force-integral-calculation-tutorial Examples: 1. Find the fluid force on a rectangular metal sheet measuring 3 feet by 4 feet that is submerged

horizontally in 6 feet of water.

53

The previous case was easy because it involved no calculus (no variation in downward force). The next case does involve calculus because the downward force varies depending on where on the vertically submerged surface you look.

Definition of Force Exerted by Fluid (Vertical Case):

�

F PA whA Where w is a constant We will use horizontal rectangles to determine the force at any depth. These rectangular areas are defined by the length of the rectangle (L) times the width (

�

dsomething). Since the rectangles must be horizontal, the length would be L(y) and the width would be dy. The height (or depth) is known, therefore, as h(y). So we have the rule:

�

F w h(y)L(y)dyc

d

³

where c represents the lowest vertical distance of the object and d represents the highest.

2. Find the fluid force on a rectangular metal sheet measuring 5 feet deep by 1 foot wide that is submerged

vertically such that the topmost point is 1 foot below the surface of the water. 3. A vertical gate in a dam has the shape of an isosceles trapezoid 8 feet across the top and 6 feet across th

e bottom, with a height of 5 feet. What is the fluid force on the gate if the top of the gate is 4 feet below the surface of the water?

4. A triangular plate, base 5 feet, height 6 feet, is submerged in water, vertex down, plane vertical, and 2 f

eet below the surface. Find the total force on one face of the plate.

54

Calculus 2 Name:____________________________________ Lesson- Inverse Trig Functions, Differentiation,

and Integration Date:_____________________________________ Objective: To learn how to find derivatives and anti-derivative of inverse trig functions You learned to use the following notation in Precalculus: 1 1cos arccos , sin arcsin ,x x x x� � etc. You also learned that the inverse trig functions are restricted so that they will be functions. In Calculus the restrictions are understood to apply without our using a capital letter for the inverse.

0 arccos

0 arcsec (but )2

0 arccot

x

x x

x

SSS

S

d d

d d z

� �

arcsin2 2

arccsc (but 0)2 2

arctan2 2

x

x x

x

S S

S S

S S

� d d

� d d z

� � �

_________________________________________________________________________________

Ex. 1arcsin2

§ ·� ¨ ¸© ¹

Ex. 1 2cos2

� § ·� ¨ ¸¨ ¸© ¹

__________________________________________________________________________________ Ex. Use your calculator to find: (a) � �arctan 0.3� (b) � �1csc 2.4� � __________________________________________________________________________________ Ex. Evaluate:

(a) 4sin arccos5

§ · ¨ ¸© ¹

(b) � �sin arccos x (c) � �sec arctan 3x

55

We can use implicit differentiation to derive the differentiation formulas for the inverse trig functions.

> @arcsind udx

> @arctand udx

> @arcsecd udx

Add to your Formula Sheet:

> @arcsind udx

> @arccosd udx

> @arctand udx

> @arccotd udx

> @arcsecd udx

> @arccscd udx

� � � �� �

Ex. arcsin 2f x x

f x

c

� � � �� �

Ex. arctan 3f x x

f x

c

� � � �� �

2Ex. arcsec xf x e

f x

c

� � � �� �� �

Ex. cos arcsin 3f x x

f x

c

56

Ex. Find an equation of the tangent line to the graph of 1 arccos2

y x at the point on the graph

where x = 22

� .

We can also go the other way and find the integral:

Ex. Evaluate: 225

dxx�³ Ex. Evaluate: 2

2 74 13x dx

x x�

� �³

Problem Set Find the derivative.

1. � � � �3arcsin 5f x x 5. � �arcsin 2x

yx

2. � �3arccos 2y x 6. � � � �2 2arctanh x x x

3. � � � �arcsec 3g x x 7. � � � �cos arcsinp x x

4. � � arctan5xf x § · ¨ ¸

© ¹ 8. � � � �sec arctanq x x

9. Given � � � �2arccos �f x x Find the line tangent to the function when x = 32

�

Find each integral:

10. 2

316x dx

x�

�³ 11. 24

dxx�³ 12.

28 2dxx x� �

³

57

Calculus 2 Name:____________________________________ Lesson- Hyperbolic Trig Functions

Date:_____________________________________ Objective: To learn the nature of Hyperbolic trig functions

Definitions:

2sinh

xx eex��

2

coshxx eex

�� xx

xx

eeeex�

�

��

tanh

Using these relationships, find the following: Find which hyperbolic trig functions are even and odd and show their relationship. 1) sinh( )T� 2) cosh( )T� 3) tanh( )T� Look at angle sum relationships. 4) sinh( )D E� 5) cosh( )D E� 4) tanh( )D E� Look at double angle relationships (hint: Use double angle identities) 7) sinh(2 )D 8) cosh(2 )D 9) tanh(2 )D

58

10) Look at the relationship between 2 2sinh ( ) cosh ( )x and x , 2 2tanh ( ) sech ( )x and x and 2 2csch ( ) coth ( )x and x

11) Can we express ex and e-x in terms of sinh x and cosh x ? 12) Find sinh -1 x = 13) Find arcosh x =

14) Find half angle results. sinh2x§ ·

¨ ¸© ¹

, cosh2x§ ·

¨ ¸© ¹

59

Calculus 2 Name:____________________________________ Lesson- Hyperbolic Trig Function Graphs, Differentiation

and Integration Date:_____________________________________ Objective: To learn how to graph Hyperbolic Trig Functions

To learn how to find derivatives and anti-derivative of Hyperbolic trig functions

Graphing Hyperbolic Trig functions and Differentiation

Graph each hyperbolic trig function then sketch the following transformations. 1) y = sinh ( 2x) 2) y = cosh (x – 3) 3) y= – tanh (x) + 2 Differentiate each hyperbolic trig function using the exponential rules and simplify. 4) sinh(x) 5) cosh(x) 6) tanh(x)

60

7) sech(x) 8) coth(x) 9) csch(x)

11) 1(cosh )ddx x

§ ·¨ ¸© ¹

12) � �(sinh cosh( ) )d xdx

Finally,… differentiate the following functions 13) sinh-1(x) 14) cosh-1(x) 15) tanh-1(x)

61

Integration of Hyperbolic trig functions. Find the integral of each hyperbolic trig functions. 1) sinh( )x dx ³ 2) cosh( )x dx ³

Integrate : 3) tanh( )x dx ³ (hint, change into sinh and cosh) 4) sec ( )h x dx ³

62

If time….

5) The equation of a catenary is ( ) cosh xf x aa

§ · ¨ ¸© ¹

A catenary is the shape a rope would form if it were

fixed at two points and allowed to hang free underneath those points. The St. Louis Arch is a catenary.

The following equations will create the graph of the St. Louis Arch but upside down:

]1)01.0[cosh(8.68)(]1)009.0[cosh(75)( � � xxgandxxf

a) Graph the system of equations that make up the arch on your calculator in a [-300,300] by [0,625]

viewing window. b) Calculate the arc length of each arch for [-300,300], average them together and find the percent error

given the actual arc length is 1480ft.