MAT37 Clculus! Lecture 7 Tody: More out Integrls (Rest of the Videos) Antiderivtives Next: Fundmentl Theorem of Clculus NEW office hours: T & R @ BA 4 officil wesite http://uoft.me/mat37 Betriz Nvrro-Lmed L6 MAT37 Jnury 7
Lower Integrl Definiton Exercise The lower integrl is the supremum of ll the lower sums. Try to write definition of the lower integrl tht s similr to the lterntive definition elow. Recll the equivlent definition of supremum we found lst clss: Definition If S is n upper ound of set A, then S is the supremum of A if it stisfies the following: ɛ >, x A such tht S ɛ < x S. Betriz Nvrro-Lmed L6 MAT37 Jnury 7
Riemnn Sums Compute ( x) dx. Betriz Nvrro-Lmed L6 MAT37 Jnury 7
Riemnn Sums Compute ( x) dx. Betriz Nvrro-Lmed L6 MAT37 Jnury 7
Riemnn Sums Compute ( x) dx. Betriz Nvrro-Lmed L6 MAT37 Jnury 7
Riemnn Sums Compute ( x) dx. Betriz Nvrro-Lmed L6 MAT37 Jnury 7
Riemnn Sums Compute ( x) dx. Betriz Nvrro-Lmed L6 MAT37 Jnury 7
Riemnn Sums Compute ( x) dx. Let P n e the prtition tht reks the intervl [, ] into n equl suintervl. Wht is x? Wht is x i? 3 Write the Riemnn Sum for x i = left endpoint of [x i, x i ]. 4 Write the Riemnn Sum for x i = right endpoint of [x i, x i ]. Betriz Nvrro-Lmed L6 MAT37 Jnury 7
Riemnn Sums Compute ( x) dx. Betriz Nvrro-Lmed L6 MAT37 Jnury 7
Integrl Interprettion s Are Exmple Evlute the following integrl: 3 (x )dx. Betriz Nvrro-Lmed L6 MAT37 Jnury 7
Integrl Interprettion s Are Exmple Evlute the following integrl: 3 (x )dx. y = f (x) A 3 x A Betriz Nvrro-Lmed L6 MAT37 Jnury 7
Properties of the Definite Integrl Let f nd g e integrle functions, nd let,, c e ny rel numers. [order of limits] f (x) dx = f (x) dx + [constnt multiple] cf (x) dx = c y = cf (x) f (x) dx y = f (x) Betriz Nvrro-Lmed L6 MAT37 y = cf (x) Jnury 7
3 [sum] (f (x) + g(x)) dx = f (x) dx + y = f (x) + g(x) g(x) dx 4 [dditivity] y = f (x) y = g(x) + = f (x) dx = c f (x) dx + c f (x) dx y = f (x) c f (x)dx f (x)dx c c Betriz Nvrro-Lmed L6 MAT37 Jnury 7
Comprison Properties of the Integrl The following properties re true only if. 5 [integrl of non-negtive function] 6 [domintion] f (x) on [, ] If f (x) g(x) on [, ] f (x) dx. f (x) dx g(x) dx. Betriz Nvrro-Lmed L6 MAT37 Jnury 7
Properties of the Definite Integrl Exmple If f (x)dx =, 5 f (x)dx = 3, find ech of the following integrls: () () 5 (f (x) + g(x)) dx f (x) dx (c) (d) g(x)dx = 5, f (x) dx g(x) dx g(x)dx =, Betriz Nvrro-Lmed L6 MAT37 Jnury 7
Antiderivtives Definition (Antiderivtive) A function F is n ntiderivtive of function f on n intervl I if F (x) = f (x) for ll x in I. Betriz Nvrro-Lmed L6 MAT37 Jnury 7
Antiderivtives Definition (Antiderivtive) A function F is n ntiderivtive of function f on n intervl I if F (x) = f (x) for ll x in I. Exmple 3 Find the ntiderivtive of f (x) = 3x. Betriz Nvrro-Lmed L6 MAT37 Jnury 7
Antiderivtives Theorem (Generl form of ntiderivtive) If F is n ntiderivtive of f on n intervl I, then the most generl ntiderivtive of f on I is where C R is n ritrry constnt. f (x)dx = F (x) + C Note: f (x)dx represents the collection of ll functions whose derivtive is f (x). Exmple 4 Find function f such tht f (x) = 3x nd f () =. Betriz Nvrro-Lmed L6 MAT37 Jnury 7
Exmple 5 Find the ntiderivtive of f (x) = (3x + 5) 7. Betriz Nvrro-Lmed L6 MAT37 Jnury 7
Exmple 5 Find the ntiderivtive of f (x) = (3x + 5) 7. Here s the generl strtegy in the form of flow digrm: guess not close check close djust check correct not quite correct write most generl ntiderivtive Betriz Nvrro-Lmed L6 MAT37 Jnury 7
Exmple 6 Find function f (x) if f (x) = sin x + e x x, nd f () =, f () =. Betriz Nvrro-Lmed L6 MAT37 Jnury 7
Exmple 7 Evlute e x e x + dx Betriz Nvrro-Lmed L6 MAT37 Jnury 7
Exmple 8 Evlute sin x dx x Betriz Nvrro-Lmed L6 MAT37 Jnury 7
The fundmentl theorem of clculus dels with functions of the form g(x) = x f (t) dt, where f is continuous function on [, ] nd x vries etween nd. For exmple, if f is non-negtive, then g(x) cn e interpreted s the re under the grph of f etween nd x, where x vries from to. You cn think of g s the re so fr function. Betriz Nvrro-Lmed L6 MAT37 Jnury 7
Are so Fr Function Let f (t) = t nd =, then the function g(x) = x tdt represents the re under the curve in the picture. Thus, g(x) x Betriz Nvrro-Lmed L6 MAT37 Jnury 7