The Area Between Two Curves Lesson 6.1 When f(x) • Consider taking the definite integral for the function shown below. a b b f (
Download ReportTranscript The Area Between Two Curves Lesson 6.1 When f(x) • Consider taking the definite integral for the function shown below. a b b f (
The Area Between Two Curves Lesson 6.1 When f(x) < 0 • Consider taking the definite integral for the function shown below. a b b f ( x)dx f(x) a • The integral gives a ___________ area • We need to think of this in a different way Another Problem • What about the area between the curve and the x-axis for y = x3 • What do you get for the integral? 2 3 x dx 2 • Since this makes no sense – we need another way to look at it Solution • We can use one of the properties of integrals ___ b f ( x)dx a ___ f ( x )dx ___ f ( x )dx ___ • We will integrate separately for _________ and __________ 2 3 x dx 2 0 2 2 0 3 3 x dx x dx 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)=x2 – 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) = x2 – 4 and g(x) = 8 – 2x2 • 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 f ( x) 9 9x g( x) 9 x x axis • Again, we must split the region into two parts • _________________ and ______________ Slicing the Shark the Other Way f ( x) 9 9x g( x) 9 x x axis • We could make these graphs as ________________ 1 j ( y ) x 9 y 2 and 9 • Now each slice is _______ by (k(y) – j(y)) k ( y) x 9 y 2 Practice • Determine the region bounded between the given curves • Find the area of the region y x y x2 6 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 x 8 y x y 2 2 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 b • Think of integrating as an accumulation of the areas of the rectangles from 0 to b ___ 0 sin x dx Integration as an Accumulation Process • We can think of this as a function of b b ______ sin x dx cos ( x) 0 cos b 1 b 0 • This gives us the accumulated area under the curve on the interval [0, b] Try It Out • Find the accumulation function for 1 2 F ( x) t 2 dt 2 0 x • Evaluate • F(0) • F(4) • F(6) Applications • The surface of a machine part is the region between the graphs of y1 = |x| and y2 = 0.08x2 +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) f(x) on the interval [a, b] a • Now consider revolving b that segment of curve about the x axis • What kind of functions generated these solids of revolution? Disks f(x) • 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? Volume of slice = ___________ dx Disks • To find the volume of the whole solid we sum the volumes of the disks • Shown as a definite integral V _____________ f(x) a b Try It Out! • Try the function y = x3 on the interval 0 < x < 2 rotated about x-axis Revolve About Line Not a Coordinate Axis • Consider the function y = 2x2 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 f(x) g(x) a • Now we have a hollow solid • We will sum the volumes of washers • As an integral b V __________ _________dx a b Application • Given two functions y = x2, and y = x3 • Revolve region between about x-axis What will be the limits of integration? 1 V _______________ dx 0 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 ____ V ______ f ( y) dy 2 ____ Flat Washer • Determine the volume of the solid generated by the region between y = x2 and y = 4x, revolved about the y-axis • Radius of inner circle? • f(y) = _____ • Radius of outer circle? • f ( y) ______ • Limits? • 0 < y < 16 Cross Sections • Consider a square at x = c with side equal to side s = f(c) a 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 ba __________ n i 1 n f(x) b Cross Sections ba f ( xi ) n i 1 n 2 f(x) • This suggests a limit and an integral a c ba lim f ( xi ) _____________ n n i 1 n 2 b Cross Sections • We could do similar summations (integrals) for other shapes • Triangles • Semi-circles • Trapezoids f(x) a c b 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 31 - 39 odd, 49, 53, 57 Volume: The Shell Method Lesson 7.3 4 3 Find the volume generated when2 this shape is revolved about the 1 y axis. 0 1 2 3 y 4 5 6 4 2 x 10 x 16 9 7 8 We can’t solve for x, so we can’t use a horizontal slice directly. 4 3 2 1 If we take a ____________slice and revolve it about the y-axis we get a cylinder. 0 1 2 3 y 4 5 6 4 2 x 10 x 16 9 7 8 Shell Method • Based on finding volume of cylindrical shells • Add these volumes to get the total volume • Dimensions of the shell • _________of the shell • _________of the shell • ________________ The Shell • Consider the shell as one of many of a solid of dx revolution f(x) f(x) – g(x) x g(x) • The volume of the solid made of the sum of the shells Try It Out! • Consider the region bounded by x = 0, y = 0, and y 8 x2 V 2 2 2 0 x 8 x 2 dx 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 2 ____ radius ________ thickness • Use integration formula b Volume 2 radius height thickness a Rotation About x-Axis • Rotate the region bounded by y = 4x and y = x2 about the x-axis thickness = _____ _______________ = y • What are the dimensions needed? • radius • height • thickness height = _________ y V 2 y y dy 4 0 16 Rotation About Non-coordinate Axis • Possible to rotate a region around any line g(x) f(x) x=a • Rely on the basic concept behind the shell method Vs 2 radius height thickness Rotation About Non-coordinate Axis • What is the radius? r g(x) f(x) a–x • What is the height? x=c x=a f(x) – g(x) • What are the limits? c<x<a • The integral: a V __________ f ( x) g ( x) dx c Try It Out • Rotate the region bounded by 4 – x2 , x = 0 and, y = 0 about the line x = 2 • Determine radius, height, limits Try It Out • Integral for the volume is 2 V 2 __________________ dx 0 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) P0 P1 Pi • •• • • • b a • That is from P0 to Pn Pn • The distance formula for each pair of points d ( Pi 1 , Pi ) What is another way of representing this? xi xi 1 f ( xi ) f ( xi 1 ) 2 2 xi yi 2 2 xi Why? Arc Length • We sum the individual lengths n L 1 f '( xi ) dx 2 i 1 • When we take a limit of the above, we get the integral L ___________________ Arc Length • Find the length of the arc of the function for 1<x<2 4 x 1 y 4 8x2 Surface Area of a Cone • Slant area of a cone A r s s h r • Slant area of frustum A 2 L L Surface Area • Suppose we rotate the f(x) from slide 2 around the x-axis • A surface is formed • A slice gives a __________ Δx P0 P1 • •• a Pi Pn • • • xi • b Δs 2 Rr S 2 s 2 _______ 1 f '( x) x 2 Surface Area • We add the cone frustum areas of all the slices • From a to b • Over entire length of the curve b S _________ 1 f '( xi ) dx 2 a b 2 ________ 1 f '( xi ) dx 2 a Surface Area • Consider the surface generated by the curve y2 = 4x for 0 < x < 8 about the x-axis y 4x Surface Area • Surface area = Limitations • We are limited by what functions we can integrate b S 2 f ( x) 1 f '( xi ) dx 2 a • 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 50 12 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 = k x F ( x) • Force required to move through i th interval, x • W = F(xi) x x a b 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 b W F ( x) dx a 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? F kx • What is F(x) the force function? 0.5 • Work done? W 30x dx 0 4.5 k 0.15 30 k F ( x ) 30 x 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 r • Draw a picture with the b coordinate system a • 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 a the top W r 2 _________ dy 0 Pumping Liquids • Suppose tank has 8 4 • r=4 • height = 8 • filled with petroleum (54.8 lb/ft3) • What is work done to pump oil over top • Disk weight? Weight _________________ ___________ • Distance moved? • Integral? Work Done by Expanding Gas • Consider a piston of radius r in a cylindrical casing as shown here • Let p = pressure in lbs/ft2 • Let V = volume of gas in ft3 • Then the work increment involved in moving the piston Δx feet is W __________________ Work Done by Expanding Gas • So the total work done is the summation of all those increments as the gas expands from V0 to V1 W V1 p dV V0 • Pressure is inversely proportional V1 to volume W so p _________ and V0 dV Work Done by Expanding Gas • A quantity of gas with initial volume of 1 cubic foot and a pressure of 2500 lbs/ft2 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 Force __________________ • Contrast of measures of mass and force System Measure of Mass Measure of Force U.S. Slug Pound International Kilogram Newton C-G-S Gram Dyne 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 10kg • 6kg Centroid • Each mass in the system has a "moment" • The product of ____________________________ from the origin 4kg 10kg 6kg • "First moment" is the __________of all the moments • The centroid is m1 x1 m2 x2 x m1 m2 Centroid • Centroid for multiple points n x m x i 1 n i i mi First moment of the system Also notated My, moment about y-axis n i 1 y • Centroid about x-axis Also notated Mx, moment about x-axis m y i 1 n i i m i 1 i Centroid • The location of the centroid is the ordered pair (x, y) x My m Mx y m • 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) 12g 7g 10g M y _______________________ M x 1______________________ m 10 7 12 x ? y ? Centroid • Consider a region under a curve of a material of uniform density • • We divide the region into a b x ____________ • 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 Centroid of Area Under a Curve • First moment with respect b to the y-axis M y x f ( x) dx a • First moment with respect b to the x-axis 1 Mx f ( x) 2 2 a • Mass of the region b m f ( x) dx a dx Centroid of Region Between Curves • Moments f(x) b M y x f ( x) g ( x) dx a b g(x) 1 2 2 M x f ( x) g ( x) dx 2a • Mass b m f ( x) g ( x) dx a Centroid x My m Mx y m Try It Out! • Find the centroid of the plane region bounded by y = x2 + 16 and the x-axis over the interval 0 < x < 4 • Mx = ? • My = ? • 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 R c Theorem of Pappus • Now revolve the region about the line L • Theorem states that the volume of the solid of revolution is V _______________ where A is the area of R L r R c 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 Pressure P ____________ • Where w is the weight-density of the liquid per unit of volume • Some example densities water 62.4 lbs/ft3 mercury 849 lbs/ft3 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 F P A wh A Fluid Force on Submerged Object • Consider a rectangular metal sheet measuring 2 x 4 feet that is submerged in 7 feet of water • Remember Pressure P w h so P = 62.4 x 7 = 436.8 • And F = P x A so F = 436.8 x 2 x 4 = 3494.4 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 d F h( y) L( y) dy c Fluid Pressure • The tank has cross section of a trapazoid • Filled to 2.5 ft with water • Water is 62.4 lbs/ft3 • Function of edge • Length of strip • Depth of strip • Integral (-4,2.5) (4,2.5) 2.5 - y (-2,0) (2,0) Assignment A • Lesson 7.7 • Page 511 • Exercises 1-25 odd