The Area Between Two Curves Lesson 6.1

When f(x) < 0 Consider taking the definite integral for the function shown below. The integral gives a ___________ area We need to think of this in a different way a b f(x)

Another Problem What about the area between the curve and the x-axis for y = x 3 What do you get for the integral? Since this makes no sense – we need another way to look at it

Solution We can use one of the properties of integrals We will integrate separately for _________ and __________ We take the absolute value for the interval which would give us a negative area.

General Solution When determining the area between a function and the x-axis Graph the function first Note the ___________of the function Split the function into portions where f(x) > 0 and f(x) < 0 Where f(x) < 0, take ______________ of the definite integral

Try This! Find the area between the function h(x)=x 2 – x – 6 and the x-axis Note that we are not given the limits of integration We must determine ________ to find limits Also must take absolute value of the integral since specified interval has f(x) < 0

Area Between Two Curves Consider the region between f(x) = x 2 – 4 and g(x) = 8 – 2x 2 Must graph to determine limits Now consider function inside integral Height of a slice is _____________ So the integral is

The Area of a Shark Fin Consider the region enclosed by Again, we must split the region into two parts _________________ and ______________

Slicing the Shark the Other Way We could make these graphs as ________________ Now each slice is _______ by (k(y) – j(y))

Practice Determine the region bounded between the given curves Find the area of the region

Horizontal Slices Given these two equations, determine the area of the region bounded by the two curves Note they are x in terms of y

Assignments A Lesson 7.1A Page 452 Exercises 1 – 45 EOO

Integration as an Accumulation Process Consider the area under the curve y = sin x Think of integrating as an accumulation of the areas of the rectangles from 0 to b b

Integration as an Accumulation Process We can think of this as a function of b This gives us the accumulated area under the curve on the interval [0, b]

Try It Out Find the accumulation function for Evaluate F(0) F(4) F(6)

Applications The surface of a machine part is the region between the graphs of y 1 = |x| and y 2 = 0.08x 2 +k Determine the value for k if the two functions are tangent to one another Find the area of the surface of the machine part

Assignments B Lesson 7.1B Page 453 Exercises 57 – 65 odd, 85, 88

Volumes – The Disk Method Lesson 7.2

Revolving a Function Consider a function f(x) on the interval [a, b] Now consider revolving that segment of curve about the x axis What kind of functions generated these solids of revolution? f(x) a b

Disks We seek ways of using integrals to determine the volume of these solids Consider a disk which is a slice of the solid What is the radius What is the thickness What then, is its volume? dx f(x)

Disks To find the volume of the whole solid we sum the volumes of the disks Shown as a definite integral f(x) a b

Try It Out! Try the function y = x 3 on the interval 0 < x < 2 rotated about x-axis

Revolve About Line Not a Coordinate Axis Consider the function y = 2x 2 and the boundary lines y = 0, x = 2 Revolve this region about the line x = 2 We need an expression for the radius _______________

Washers Consider the area between two functions rotated about the axis Now we have a hollow solid We will sum the volumes of washers As an integral f(x) a b g(x)

Application Given two functions y = x 2, and y = x 3 Revolve region between about x-axis What will be the limits of integration?

Revolving About y-Axis Also possible to revolve a function about the y-axis Make a disk or a washer to be ______________ Consider revolving a parabola about the y-axis How to represent the radius? What is the thickness of the disk?

Revolving About y-Axis Must consider curve as x = f(y) Radius ____________ Slice is dy thick Volume of the solid rotated about y-axis

Flat Washer Determine the volume of the solid generated by the region between y = x 2 and y = 4x, revolved about the y-axis Radius of inner circle? f(y) = _____ Radius of outer circle? Limits? 0 < y < 16

Cross Sections Consider a square at x = c with side equal to side s = f(c) Now let this be a thin slab with thickness Δx What is the volume of the slab? Now sum the volumes of all such slabs c f(x) b a

Cross Sections This suggests a limit and an integral c f(x) b a

Cross Sections We could do similar summations (integrals) for other shapes Triangles Semi-circles Trapezoids c f(x) b a

Try It Out Consider the region bounded above by y = cos x below by y = sin x on the left by the y-axis Now let there be slices of equilateral triangles erected on each cross section perpendicular to the x-axis Find the volume

Assignment Lesson 7.2A Page 463 Exercises 1 – 29 odd Lesson 7.2B Page 464 Exercises odd, 49, 53, 57

Volume: The Shell Method Lesson 7.3

Find the volume generated when this shape is revolved about the y axis. We can’t solve for x, so we can’t use a horizontal slice directly.

If we take a ____________slice and revolve it about the y-axis we get a cylinder.

Shell Method Based on finding volume of cylindrical shells Add these volumes to get the total volume Dimensions of the shell _________of the shell ________________

The Shell Consider the shell as one of many of a solid of revolution The volume of the solid made of the sum of the shells f(x) g(x) x f(x) – g(x) dx

Try It Out! Consider the region bounded by x = 0, y = 0, and

Hints for Shell Method Sketch the __________over the limits of integration Draw a typical __________parallel to the axis of revolution Determine radius, height, thickness of shell Volume of typical shell Use integration formula

Rotation About x-Axis Rotate the region bounded by y = 4x and y = x 2 about the x-axis What are the dimensions needed? radius height thickness _______________ = y thickness = _____

Rotation About Non-coordinate Axis Possible to rotate a region around any line Rely on the basic concept behind the shell method x = a f(x) g(x)

Rotation About Non-coordinate Axis What is the radius? What is the height? What are the limits? The integral: x = a f(x) g(x) a – x f(x) – g(x) x = c r c < x < a

Try It Out Rotate the region bounded by 4 – x 2, x = 0 and, y = 0 about the line x = 2 Determine radius, height, limits

Try It Out Integral for the volume is

Assignment Lesson 7.3 Page 472 Exercises 1 – 25 odd Lesson 7.3B Page 472 Exercises 27, 29, 35, 37, 41, 43, 55

Arc Length and Surfaces of Revolution Lesson 7.4

Arc Length We seek the distance along the curve from f(a) to f(b) That is from P 0 to P n The distance formula for each pair of points a b P0P0 P1P1 PnPn PiPi Why? What is another way of representing this?

Arc Length We sum the individual lengths When we take a limit of the above, we get the integral

Arc Length Find the length of the arc of the function for 1 < x < 2

Surface Area of a Cone Slant area of a cone Slant area of frustum s r h L

Surface Area Suppose we rotate the f(x) from slide 2 around the x-axis A surface is formed A slice gives a __________ a b P0P0 P1P1 PnPn PiPi xixi ΔsΔs ΔxΔx

Surface Area We add the cone frustum areas of all the slices From a to b Over entire length of the curve

Surface Area Consider the surface generated by the curve y 2 = 4x for 0 < x < 8 about the x-axis

Surface Area Surface area =

Limitations We are limited by what functions we can integrate Integration of the above expression is not _________________________ We will come back to applications of arc length and surface area as new integration techniques are learned

Assignment Lesson 7.4 Page 383 Exercises 1 – 29 odd also 37 and 55,

Work Lesson 7.5

Work Definition The product of The ____________exerted on an object The _______________the object is moved by the force When a force of 50 lbs is exerted to move an object 12 ft. 600 ft. lbs. of work is done ft

Hooke's Law Consider the work done to stretch a spring Force required is proportional to _________ When k is constant of proportionality Force to move dist x = Force required to move through i th interval,  x  W = F(x i )  x a b xx

Hooke's Law We sum those values using the definite integral The work done by a ____________force F(x) Directed along the x-axis From x = a to x = b

Hooke's Law A spring is stretched 15 cm by a force of 4.5 N How much work is needed to stretch the spring 50 cm? What is F(x) the force function? Work done?

Winding Cable Consider a cable being wound up by a winch Cable is 50 ft long 2 lb/ft How much work to wind in 20 ft? Think about winding in  y amt y units from the top  50 – y ft hanging dist =  y force required (weight) =2(50 – y)

Pumping Liquids Consider the work needed to pump a liquid into or out of a tank Basic concept: Work = weight x _____________ For each  V of liquid Determine __________ Determine dist moved Take summation (__________________)

Pumping Liquids – Guidelines Draw a picture with the coordinate system Determine _______of thin horizontal slab of liquid Find expression for work needed to lift this slab to its destination Integrate expression from bottom of liquid to the top a b r

Pumping Liquids Suppose tank has r = 4 height = 8 filled with petroleum (54.8 lb/ft 3 ) What is work done to pump oil over top Disk weight? Distance moved? Integral? 8 4 ___________

Work Done by Expanding Gas Consider a piston of radius r in a cylindrical casing as shown here Let p = pressure in lbs/ft 2 Let V = volume of gas in ft 3 Then the work increment involved in moving the piston Δx feet is

Work Done by Expanding Gas So the total work done is the summation of all those increments as the gas expands from V 0 to V 1 Pressure is inversely proportional to volume so p _________ and

Work Done by Expanding Gas A quantity of gas with initial volume of 1 cubic foot and a pressure of 2500 lbs/ft 2 expands to a volume of 3 cubit feet. How much work was done?

Assignment A Lesson 7.5 Page 405 Exercises 1 – 41 EOO

Moments, Center of Mass, Centroids Lesson 7.6

Mass Definition: mass is a measure of a body's ____________to changes in motion It is ___________ a particular gravitational system However, mass is sometimes equated with __________ (which is not technically correct) Weight is a type of ___________… dependent on gravity

Mass The relationship is Contrast of measures of mass and force SystemMeasure of Mass Measure of Force U.S.SlugPound InternationalKilogramNewton C-G-SGramDyne

Centroid Center of mass for a system The point where all the mass seems to be concentrated If the mass is of constant density this point is called the __________________ 4kg 6kg 10kg

Centroid Each mass in the system has a "moment" The product of ____________________________ from the origin "First moment" is the __________of all the moments The centroid is 4kg 6kg 10kg

Centroid Centroid for multiple points Centroid about x-axis First moment of the system Also notated M y, moment about y-axis Also notated M x, moment about x-axis

Centroid The location of the centroid is the ordered pair Consider a system with 10g at (2,-1), 7g at (4, 3), and 12g at (-5,2) What is the center of mass?

Centroid Given 10g at (2,-1), 7g at (4, 3), and 12g at (- 5,2) 10g 7g 12g

Centroid Consider a region under a curve of a material of uniform density We divide the region into ____________ Mass of each considered to be centered at _______________________center Mass of each is the product of the density, ρ and the area We sum the products of distance and mass a b

Centroid of Area Under a Curve First moment with respect to the y-axis First moment with respect to the x-axis Mass of the region

Centroid of Region Between Curves Moments Mass f(x) g(x) Centroid

Try It Out! Find the centroid of the plane region bounded by y = x and the x-axis over the interval 0 < x < 4 M x = ? M y = ? m = ?

Theorem of Pappus Given a region, R, in the plane and L a line in the same plane and not intersecting R. Let c be the centroid and r be the distance from L to the centroid L R c r

Theorem of Pappus Now revolve the region about the line L Theorem states that the volume of the solid of revolution is where A is the area of R L R c r

Assignment Lesson 7.6 Page 504 Exercises 1 – 41 EOO also 49

Fluid Pressure and Fluid Force Lesson 7.7

Fluid Pressure Definition: The pressure on an object at depth h is Where w is the weight-density of the liquid per unit of volume Some example densities water 62.4 lbs/ft 3 mercury849 lbs/ft 3

Fluid Pressure Pascal's Principle: pressure exerted by a fluid at depth h is transmitted _______in all __________________ Fluid pressure given in terms of force per unit area

Fluid Force on Submerged Object Consider a rectangular metal sheet measuring 2 x 4 feet that is submerged in 7 feet of water Remember so P = 62.4 x 7 = And F = P x A so F = x 2 x 4 = lbs

Fluid Pressure Consider the force of fluid against the side surface of the container Pressure at a point Density x g x depth Force for a horizontal slice Density x g x depth x Area Total force

Fluid Pressure The tank has cross section of a trapazoid Filled to 2.5 ft with water Water is 62.4 lbs/ft 3 Function of edge Length of strip Depth of strip Integral (-2,0) (2,0) (-4,2.5) (4,2.5) y

Assignment A Lesson 7.7 Page 511 Exercises 1-25 odd