Calculus 2 UNIT 1 Integration Applications of Definite Integrals LEARNING OUTCOMES After this unit students are expected to Resources The Definite Integral MIT Opencourseware https ocw mit edu resources res 18 006 calculus revisited single variable calculus fall 2010 part iv the definite integral lecture 1 the definite integral I Area and Estimating with Finite Sums The estimation of the area under the graph on the interval of the function using finite approximation can be found using 1 2 1 1 where and 1 Given is a function 2 2 Use finite approximations to estimate the area under the graph on the interval 1 4 of the function using A left hand sum with six rectangles of equal width A left hand sum with three rectangles of equal width 1 Calculus 2 UNIT 1 Integration Applications of Definite Integrals An right hand sum with six rectangles of equal width An right hand sum with three rectangles of equal width The midpoint rule with six rectangles of equal width The midpoint rule with three rectangles of equal width Note Check your work using https www desmos com calculator tgyr42ezjq 2 Calculus 2 UNIT 1 Integration Applications of Definite Integrals 2 Oil leaked from a tank at a rate of litters per hour The rate decreased as time passed and values of the rate at two hour time intervals are shown in the table hours Liter hours 0 8 7 2 7 6 4 6 8 6 6 2 8 5 7 10 5 3 a Find the lower estimates for the total amount of oil that leaked out b Find the upper estimates for the total amount of oil that leaked out 3 Calculus 2 UNIT 1 Integration Applications of Definite Integrals II Sigma Notation Limits of Finite Sums 1 Write the sums without sigma notation a b c 2 1 6 1 4 cos 1 1 1 1 2 Algebra Rules for Finite Sums 1 1 Sum Rule Constant Multiple Rule 1 Difference Rule Constant Value Rule 1 1 any number 1 1 1 1 is any constant value 2 Suppose that 1 5 and 1 6 Find the values of a 1 6 b 2 1 4 Calculus 2 UNIT 1 Integration Applications of Definite Integrals Special Summation Formulas 1 is a constant 2 1 1 2 1 6 1 1 2 3 1 1 2 4 3 Use the summation formula to evalute the following sums b c a 100 25 1 50 2 5 1 15 1 5 Calculus 2 UNIT 1 Integration Applications of Definite Integrals III The Definite Integral Definition of definite integral Let be defined on If lim 1 lim 1 2 exists for all choices of representative points 1 2 in the subintervals of f then the limit is called the definite integral of from to and is equal width denoted by Thus 1 lim lim 1 2 The number is the lower limit of integration and the number is the upper limit of integration 1 Use the de nition of the definite integral to evaluate the integral use the right endpoints to calculate the Riemann Sums 5 a b 1 3 1 2 2 2 0 c 2 3 1 Solution a Given 5 1 6 Since 1 and 5 so Let 1 3 Then 1 2 2 3 so 6 1 Since 1 an Therefore 6 1 3 1 18 18 1 3 2 Using the definition of integral lim 1 then 5 1 1 3 lim 2 1 18 6 lim 2 1 18 6 lim 1 12 108 2 1 3 lim 1 3 1 12 1 108 2 12 lim 1 1 108 2 1 1 3 lim 12 108 2 1 2 1 3 lim 12 54 1 3 lim 12 54 1 2 2 1 1 3 12 54 5 1 5 1 5 1 5 1 5 1 So 5 1 6 Calculus 2 UNIT 1 Integration Applications of Definite Integrals 7 Calculus 2 UNIT 1 Integration Applications of Definite Integrals Properties of Integral 9 1 9 1 a 2 Suppose that an are integrable and that 1 5 4 Use the properties of integral to find b 9 2 9 7 c 7 1 9 7 7 8 Calculus 2 UNIT 1 Integration Applications of Definite Integrals The Area under a Nonnegative Integral Function If is nonnegative and integrable over a closed interval then the area under the curve over is the integral of from to 3 Sketch the given functions and find the area of the region between the given curve and the axis on the interval 0 a 2 b 2 c 1 2 9 Calculus 2 UNIT 1 Integration Applications of Definite Integrals IV The Fundamental Theorem of Calculus The mean value theorem of a definite integral If f is integratable on then its average value on also called its mean is av 1 1 Graph the function and find its average value over the given interval a 2 1 on 0 3 b on 1 1 2 A Gateway Arch is 630 ft high and has a 630 ft base Its shape can be modelled by the parabola 630 1 2 315 Find the average height of the arch above the ground 10 Calculus 2 UNIT 1 Integration Applications of Definite Integrals The fundamental theorem of Calculus If f is continuous on then on and its derivative is is continuous on and differentiable 3 a c 3 1 2 1 cos b d 3 sin 1 2 5 4 1 3 2 The fundamental theorem of Calculus the evaluation theorem Let be continuous on Then Where is any antiderivative of such that is 3 Evaluate the following integrals 4 8 7 0 4 4 2 2 cos 1 b d f 9 1 2 2 2 3 1 ln 8 0 a c e 11 Calculus 2 UNIT 1 Integration Applications of Definite Integrals V The Substitution Rule for Integrals The substitution rule for indefinite integrals If is a differentiable function whose range is an interval and is continuous on then The substitution rule for definite integrals If is continuous on and is continuous on the range of then 1 Evaluate the following integrals 8 cos 4 2 3 2 2 1 4 sin3 cos 2 2 3 1 1 1 4 0 1 16 2 b d f h 6 1 3 2 4 0 ln 4 0 sin cos3 3 2 a c e g 12 Calculus 2 UNIT 1 Integration Applications of Definite Integrals VI The Area Between Curves 1 a Use definite integral to find the area between the curves b Larson Edwards 2015 13 Calculus 2 UNIT 1 Integration Applications of Definite Integrals 2 Find the area of the region by integrating With respect to With respect to Compare your results Which method is simpler In general will this method always be simpler than the other one Why or why not i ii iii a b 14 Calculus 2 UNIT 1 Integration Applications of Definite Integrals Sketch the region bounded by the graphs of the functions and find the area of the region 2 1 …