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Unit 5IntegrationSection 1Antiderivatives and Indefinite IntegrationAntiderivative: a function F for which __________________ for all x in I.For example, to find a function F whose derivative is f(x)=3x2, you could conclude that F(x)=x3 because F'(x)=3x2Note: F is AN antiderivative, not THE antiderivative.Theorem: Representation of AntiderivativesIf F is an antiderivative of f on an interval I, then G is an antiderivative of f on the interval I if and only if G is of the form _________________________, for all x in I where C is a constant.Constant of Integration: ____________________ in the family of derivatives G(x)=F(x)+CGeneral Antiderivative: the family of functions represented by G where _____________________General Solution: ____________________Differential Equation (in x and y): an equation that involves __________________________________Example 1.1Find the general solution of the differential equation y'=2.Solving Differential EquationsWrite in differential form: dy=f(x)dxAntidifferentiation (Indefinite Integration): the operation of finding all solutions of a differential equation denoted by ∫-22860016383000∫f(x)dx: dx: Basic Integration Rules-685800240030000-685800-45720000Example 1.2Find the general solution of ∫3xdxExample 1.3Rewrite before integrating:A. ∫1/x3(dx)B. ∫√x(dx)C. ∫2sin(x)(dx)Example 1.4A. ∫dxB. ∫(x+2)dxC. ∫(3x4-5x2+x)dx508023368000Example 1.5508023431500Example 1.6Example 1.7A. ∫2/√x(dx)B. ∫(t2+1)2dtC. ∫(x3+3)/x(dx)D. ∫?x(x-4)(dx)Particular Solution: an antiderivative in which you know the _________________________Initial Condition: the value of _____________ for one value of xExample 1.8Find the general solution of F(x)=1/x2, x>0.Then find the particular solution that satisfies the initial condition F(1)=0Example 1.9A ball is thrown upward with an initial velocity of 64 ft/sec from an initial height of 80 ft.Find the position function giving the height s as a function of the time t.b. When does the ball hit the ground?Section 2AreaSigma Notation: the sum of n terms a1, a2, a3, ..., an is written aswhere i is the index of summation, ai is the ith term of the sum, and the upper and lower bounds of summation are n and 1.Example 2.101587500Properties of SummationSummation FormulasExample 2.2262890011430000AreaA=bh is the definition of the area of a rectangleExhaustion Method: to find the area of conics (curved figures). Squeeze the area between an inscribed polygon and a circumscribed oneExample 2.357150065532000320040065532000Use 5 rectangles to find 2 approximations of the area lying between the graph of f(x)=-x2+5 and the x-axis between x=0 and x=2.Upper and Lower SumsGiven a plane region bounded above by the graph of a nonnegative continuous function y=f(x), bounded below by the x-axis, and bounded by the vertical lines x=a and x=b, to approximate the area, subdivide the interval [a,b] into n subintervals of width ?x=(b-a)/n. The Extreme Value Theorem guarantees the existence of a maximum and minimum in each subinterval.f(mi)=minimum value of f(x) in the ith subintervalf(Mi)=maximum value of f(x) in the ith subintervalInscribed Rectangle: lies inside the subregionCircumscribed Rectangle: extends outside the subregionLower Sum: the sum of the areas of the inscribed rectanglesUpper Sum: the sum of the areas of the circumscribed rectangles-22860014287500Example 2.4508046799500Find the upper and lower sums for the region bounded by the graph of f(x)=x2 and the x-axis between x=0 and x=2.5918209906000-45720024701500-800735-45720000Example 2.5Find the area of the region bounded by the graph of f(x)=x3, the x-axis, and the vertical lines x=0 and x=1.013271500Example 2.6 Find the area of the region bounded by the graph of f(x)=4-x2, the x-axis, and the vertical lines x=1 and x=2.Example 2.7Find the area of the region bounded by the graph of f(y)=y2 and the y-axis for 0≤y≤1.Midpoint RuleUse the midpoint of the interval to approximate the area.Example 2.8Use the Midpoint Rule with n=4 to approximate the area of the region bounded by the graph of f(x)=sinx and the x-axis for 0≤x≤π.Section 3Riemann Sums and Definite Integrals445770021844000Example 3.1508046799500Consider the region bounded by the graph of f(x)=√x and the x-axis for 0≤x≤1. Evaluate the limitwhere ci is the right endpoint of the partition given by ci=i2/n2 and ?xi is the width of the ith interval.-34290083312000Norm (of a partition): the width of the largest subinterval of a partition ?, denoted by ||?||035814000Regular (partition): every subinterval is of equal width092456000-443230045212000Theorem: Continuity Implies IntegrabilityIf a function f is continuous on the closed interval [a,b], then f is ________________ on [a,b]. That is, _____________ exists.Example 3.2Evaluate the definite integral-212xdxTheorem: The Definite Integral as the Area of a RegionIf f is continuous and nonnegative on the closed interval [a,b], then the area of the region bounded by the graph of f, the x-axis, and the vertical lines x=a and x=b is2971800-63500011366500Example 3.3Sketch the region corresponding to each definite integral. Then evaluate each integral using a geometric formula.a. 134dxb. 03x+2dxc. -224-x2dxDefinitions of Two Special Definite Integrals1. If f is defined at x=a, then2. If f is integrable on [a,b], thenExample 3.4Evaluate:a. ππsinxdxb. 30x+2dxTheorem: Additive Interval PropertyIf f is integrable on the three closed intervals determined by a, b, and c, thenExample 3.5 -11|x|dxTheorem: Properties of Definite IntegralsIf f and g are integrable on [a,b] and k is a constant, then the functions kf and f±g are integrable on [a,b], and1.2.Example 3.6Evaluate 13-x2+4x-3dx using each of the following values. 13x2dx=263 13xdx=4 13dx=2Theorem: Preservation of Inequality1. If f is integrable and nonnegative on the closed interval [a,b], then2. If f and g are integrable on the closed interval [a,b] and f(x)≤g(x) for every x in [a,b], thenSection 4The Fundamental Theorem of CalculusTheorem: The Fundamental Theorem of CalculusIf a function f is continuous on the closed interval [a,b] and F is an antiderivative of f on the interval [a,b], then-5715007366000Example 4.1a. 12x2-3dxb. 143xdxc. 0π/4sec2xdxExample 4.2 02|2x-1|dxExample 4.3Find the area of the region bounded by the graph of y=2x2-3x+2, the x-axis, and the vertical lines x=0 and x=2.45720002032000Theorem: Mean Value Theorem for IntegralsIf f is continuous on the closed interval [a,b], then there exists a number c in the closed interval [a,b] such thatAverage Value of a Function on an IntervalIf f is integrable on the closed interval [a,b], then the average value of f on the interval isExample 4.4Find the average value of f(x)=3x2-2x on the interval [1,4]Example 4.5At different altitudes in Earth's atmosphere, sound travels at different speeds. The speed of sound s(x) (in meters per second) can be modeled by-4x+341,0≤x<11.5295,11.5≤x<22s(x)=3/4x+278.5,22≤x<323/2x+254.5,32≤x<50-3/2x+404.5,50≤x≤80where x is the altitude in kilometers. What is the average speed of sound over the interval [0,80]?-45720034290000The Second Fundamental Theorem of CalculusExample 4.6Evaluate the function Fx=0xcostdtat x=0, π/6, π/4, π/3, and π/2Theorem: The Second Fundamental Theorem of CalculusIf f is continuous on an open interval I containing a, then, for every x in the interval,Example 4.7Evaluate ddx[0xt2+1dt]Example 4.8Find the derivative of Fx=π/2x3costdtTheorem: The Net Change TheoremThe definite integral of the rate of change of quantity F’(x) gives the total change, or net change, in that quantity on the interval [a,b]Example 4.9A chemical flows into a storage tank at a rate of (180+3t) liters per minute, where t is the time in minutes and 0≤t≤60. Find the amount of the chemical that flows into the tank during the first 20 minutes.Displacement: net change in position Total Distance Traveled: since velocity can be negativeExample 4.10The velocity (in feet per second) of a particle moving along a line is v(t)=t3-10t2+29t-20 where t is the time in seconds.a. What is the displacement of the particle on the time interval 1≤t≤5?b. What is the total distance traveled by the particle on the time interval 1≤t≤5?Section 5Integration by SubstitutionTheorem: Antidifferentiation of a Composite FunctionLet g be a function whose range is an interval I, and let f be a function that is continuous on I. If g is differentiable on its domain and F is an antiderivative of f on I, thenLetting u=g(x) gives du=g’(x)dx andExample 5.1Find ∫(x2+1)2(2x)dxExample 5.2Find ∫5cos(5x)dxConstant Multiple RuleExample 5.3Find the indefinite integral∫x(x2+1)2dxChange of VariablesExample 5.4∫√2x-1dxExample 5.5∫x√2x-1dxExample 5.6∫sin2(3x)cos(3x)dx-571500-34290000Theorem: The General Power Rule for IntegrationIf g is a differentiable function of x, thenEquivalently, if u=g(x), thenExample 5.7a. ∫3(3x-1)4dxb. ∫(2x+1)(x2+x)dxc. ∫3x2√x3-2dxd. ∫(-4x)/(1-2x2)2dxe. ∫cos2xsinxdxTheorem: Change of Variables for Definite IntegralsIf the function u=g(x) has a continuous derivative on the closed interval [a,b] and f is continuous on the range of g, thenExample 5.8 01x(x2+1)3dxExample 5.9 15x2x-1dxTheorem: Integration of Even and Odd FunctionsLet f integrable on the closed interval [a,-a]. 1. If f is an even function, then2. If f is an odd function, thenExample 5.10 -π2π2sin3xcosx+sinxcosxdxSection 6Numerical IntegrationTheorem: The Trapezoidal RuleLet f be continuous on [a,b]. The Trapezoidal Rule for approximating abfxdx isMoreover, as n→∞, the right-hand side approaches abfxdx.Example 6.1Use the Trapezoidal Rule to approximate 0πsinxdxCompare the results when n=4 and n=8.Theorem: Integral of px=Ax2+Bx+CIf px=Ax2+Bx+C, thenTheorem: Simpson’s RuleLet f be continuous on [a,b] and let n be an even integer. Simpson’s Rule for approximating abfxdx isMoreover, as n→∞, the right-hand side approaches abfxdx.Example 6.2Use Simpson's Rule to approximate 0πsinxdxCompare the results for n=4 and n=8 ................
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