Calculating the Value of a Definite Integral

Transcription

1 Definition of the Definite Integrl Definition of the Definite Integrl If the function f is continuous on the intervl [, ], then the definite integrl of f over the intervl [, ] is f (x) dx = lim n n f (x i ) x i=1 where, s usul x =, nd x i is ny numer in the suintervl n [x i 1, x i ]. The function f is the integrnd nd the numers nd re the (lower nd upper) limits of integrtion This definition mkes sense, since the vlue of the limit of Riemnn sums s n is independent of the choice for smple points, even though different choices for the smple points give different vlues for the Riemnn sums for the sme vlue of n. Clint Lee Mth 122 Lecture 4: The Definite Integrl 2/13 Clculting the Vlue of Definite Integrl The Vlue of Definite Integrl The vlue of definite integrl cn e clculted Exctly y evluting Riemnn sums for the integrl using specific choice for smple points to otin n expression in terms of n, nd then evluting the limit s n. Exctly using the Fundmentl Theorem of Clculus (s developed in future lectures). Approximtely using Riemnn sum for specific, finite, vlue of n nd choice for smple points. Clint Lee Mth 122 Lecture 4: The Definite Integrl 3/13

2 Over nd Under Estimtes from Riemnn Sums for Monotonic Functions Riemnn Sums for Monotonic Functions If the function f is incresing on the intervl [, ], then the Riemnn sum using right-hnd endpoints gives n overestimte of the definite integrl, nd the Riemnn sum using left-hnd endpoints gives n underestimte of the definite integrl If the function f is decresing on the intervl [, ], then the Riemnn sum using right-hnd endpoints gives n underestimte of the definite integrl, nd the Riemnn sum using left-hnd endpoints gives n overestimte of the definite integrl Clint Lee Mth 122 Lecture 4: The Definite Integrl 4/13 Applictions of the Definite Integrl The Definite Integrl s n Are, Distnce, Totl Chnge If f is continuous nd positive on the intervl [, ], then f (x) dx = re of region S ounded y y = f (x) over the intervl [, ] f (t) dt = totl distnce trveled t velocity v = f (t) over the time intervl [, ] f (t) dt = totl chnge when rte of chnge is r = f (t) over the time intervl [, ] Clint Lee Mth 122 Lecture 4: The Definite Integrl 5/13

3 Net Are Note tht the definition of the definite integrl does not require tht the function f is positive over the intervl of integrtion [, ]. If the function f tkes negtive vlues over the intervl [, ], some terms in the Riemnn sum for the definite integrl re negtive. For ny pproximting rectngle for which the smple point is in n intervl where f (x) < 0, the re is x x i f (x i ) x < 0 Clint Lee Mth 122 Lecture 4: The Definite Integrl 6/13 Net Are Hence, the definite integrl counts ny re ove the x-xis s positive, nd ny re elow the x-xis s negtive. + + So tht f (x) dx = { Net Are etween the grph of f nd the x- xis, counting re ove the x-xis positive nd re elow the x-xis negtive Clint Lee Mth 122 Lecture 4: The Definite Integrl 7/13

4 Displcement & Net Chnge Displcement & Net Chnge In the sme wy s for the Net Are, if the function f tht gives the velocity of moving oject or the rte of chnge of quntity Q tkes on negtive vlues over the intervl [, ], then f (t) dt = Displcement of the oject over the time intervl [, ] = s () s () f (t) dt = Net Chnge in the quntity Q over the time intervl [, ] = Q () Q () Clint Lee Mth 122 Lecture 4: The Definite Integrl 8/13 Bsic Properties of the Definite Integrl Bsic Properties of the Definite Integrl If the function f is continuous on the intervl [, ] nd c is constnt, then c dx = c ( ) cf (x) dx = c [f (x) + g (x)] dx = [f (x) g (x)] dx = f (x) dx f (x) dx + f (x) dx Ech of these properties cn e proved using the Riemnn sum definition of the definite integrl. Clint Lee Mth 122 Lecture 4: The Definite Integrl 9/13

5 The Difference Property The Difference Property [f (x) g (x)] dx = f (x) dx cn e interpreted in terms of res. Note tht f (x) dx = re under y = f (x) over [, ] = re under y = g (x) over [, ], nd { re etween f (x) dx = y = f (x) nd y = g (x) over [, ] f (x) g (x) Clint Lee Mth 122 Lecture 4: The Definite Integrl 10/13 The Difference Property The Difference Property mens tht the re etween two curves cn e computed either y computing the two res seprtely nd sutrcting, or y integrting the difference etween the functions defining the two ounding curves Clint Lee Mth 122 Lecture 4: The Definite Integrl 11/13

6 Properties Involving Limits of Integrtion Properties Involving Limits of Integrtion If the function f is continuous on the intervl [, ] nd c is constnt, then c dx = 0 f (x) dx = f (x) = c f (x) dx f (x) dx + c f (x) dx The lst property ove does not require tht c is etween nd. Clint Lee Mth 122 Lecture 4: The Definite Integrl 12/13 Comprison Properties Comprison Properties If the functions f nd g re continuous on the intervl [, ], nd m nd M re constnts, then f (x) 0 f (x) g (x) f (x) dx 0 f (x) dx If m f (x) M for x in [, ], then m ( ) f (x) dx M ( ) Clint Lee Mth 122 Lecture 4: The Definite Integrl 13/13