Math 190 Chapter 5 Lecture Notes. Professor Miguel Ornelas


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1 Mth 19 Chpter 5 Lecture Notes Professor Miguel Ornels 1
2 M. Ornels Mth 19 Lecture Notes Section 5.1 Section 5.1 Ares nd Distnce Definition The re A of the region S tht lies under the grph of the continuous function f is the limit of the sum of res of pproximting rectngles: A = lim n R n = lim n [f(x 1) x + f(x 2 ) x + + f(x n ) x] Estimte the re under the grph of f using four pproximting rectngles nd tking the smple points to be (i) right endpoints nd (ii) midpoints. In ech cse sketch the curve nd the rectngles. f(x) = x 2 ln x 1 x 5 Section 5.1 continued on next pge... 2
3 M. Ornels Mth 19 Lecture Notes Section 5.1 (continued) The velocity grph of cr ccelerting from rest to speed of 12 km/h over period of 3 seconds is shown. Estimte the distnce trveled during this period. 3
4 M. Ornels Mth 19 Lecture Notes Section 5.2 Section 5.2 The Definite Integrl Definition of Definite Integrl Midpoint Rule where nd f(x) dx f(x) dx = lim n i=1 n f (x i ) x n f ( x i ) x = x [f ( x 1 ) + + f ( x n )] i=1 x = b n x i = 1 2 (x i 1 + x i ) = midpoint of [x i 1, x i ] Properties of the Integrl f(x) dx = b f(x) dx = f(x) dx 3. c dx = c(b ), where c is ny constnt 4. [f(x) + g(x)] dx = f(x) dx + g(x) dx 5. cf(x) dx = c f(x) dx, where c is ny constnt 6. [f(x) g(x)] dx = f(x) dx g(x) dx 7. c f(x) dx + f(x) dx = c f(x) dx 8. If f(x) for x b, then f(x) dx 9. If f(x) g(x) for x b, then f(x) dx g(x) dx Section 5.2 continued on next pge... 4
5 M. Ornels Mth 19 Lecture Notes Section 5.2 (continued) 1. If m f(x) M for x b, then m(b ) f(x) dx M(b ) Use the Midpoint Rule with the given vlue of n to pproximte the integrl. Round the nswer to four deciml plces. 1 x3 + 1 dx, n = 5 Express the limit s definite integrl on the given intervl. lim n i=1 n x i 1 + x 3 i x, [2, 5] Section 5.2 continued on next pge... 5
6 M. Ornels Mth 19 Lecture Notes Section 5.2 (continued) The grph of g consists of two stright lines nd semicircle. Use it to evlute ech integrl. (.) 2 g(x) dx (b.) 6 g(x) dx (c.) 7 2 g(x) dx Evlute the integrl by interpreting it in terms of res. 9 ( ) 1 3 x 2 dx Section 5.2 continued on next pge... 6
7 M. Ornels Mth 19 Lecture Notes Section 5.2 (continued) If 8 f(x) dx = 7.3 nd 4 f(x) dx = 5.9, find f(x) dx. Find 5 { 3 for x < 3 f(x) dx if f(x) = x for x 3 Use the properties of integrls to verify the inequlity without evluting the integrl x2 dx 1 + x dx Section 5.2 continued on next pge... 7
8 M. Ornels Mth 19 Lecture Notes Section 5.2 (continued) Use property 1 to estimte the vlue of the integrl. 2 (x 3 3x + 3) dx 8
9 M. Ornels Mth 19 Lecture Notes Section 5.3 Section 5.3 The Fundmentl Theorem of Clculus The Fundmentl Theorem of Clculus Suppose f is continuous on [, b]. 1. If g(x) = x f(t) dt, then g (x) = f(x). 2. f(x) dx = F (b) F (), where F is ny ntiderivtive of f, tht is, F = f. Use Prt 1 of the Fundmentl Theorem of Clculus to find the derivtive of the function.. g(x) = x 1 ln(1 + t 2 ) dt b. y = x 4 cos 2 θ dθ Evlute the integrl.. 1 (1 8v v 7 ) dv b. 4 (4 t) t dt Section 5.3 continued on next pge... 9
10 M. Ornels Mth 19 Lecture Notes Section 5.3 (continued) c. 3 (2 sin x e x ) dx d. 3 1 y 3 2y 2 y y 2 dy Sketch the region enclosed by the given curves nd clculte its re. y = x 3, y =, x = 1 1
11 M. Ornels Mth 19 Lecture Notes Section 5.4 Section 5.4 Indefinite Integrls nd the Net Chnge Theorem Tble of Indefinite Integrls x n dx = xn+1 n C 1 x dx = ln x + C e x dx = e x + C b x dx = bx ln b + C sin x dx = cos x + C cos x dx = sin x + C sec 2 x dx = tn x + C csc 2 x dx = cot x + C sec x tn x dx = sec x + C csc x cot x dx = csc x + C 1 x dx = tn 1 x + C 1 1 x 2 dx = sin 1 x + C sinh x dx = cosh x + C cosh x dx = sinh x + C Find the generl indefinite integrl. t(t t + 2) dt b. sec t(sec t + tn t) dt Section 5.4 continued on next pge... 11
12 M. Ornels Mth 19 Lecture Notes Section 5.4 (continued) Evlute the Integrl ( 1 x 2 4 ) x 3 dx b p 2 dp c. 2 1 (x 1) 3 x 2 dx d. 2 2x 1 dx The velocity function (in meters per second) is given for prticle moving long line. Find () the displcement nd (b) the distnce trveled by the prticle during the given time intervl. v(t) = t 2 2t 3, 2 t 4 12
13 M. Ornels Mth 19 Lecture Notes Section 5.5 Section 5.5 The Substitution Rule The Substitution Rule If u = g(x) is differentible function whose rnge is n intervl I nd f is continuous on I, then f(g(x))g (x) dx = f(u) du The Substitution Rule for Definite Integrls If g is continuous on [, b] nd f is continuous on the rnge of u = g(x), then f(g(x))g (x) dx = g(b) g() f(u) du Evlute the indefinite integrl.. x 2 e x3 dx b. sin x sin(cos x) dx c. x x dx d. x 3 x dx Section 5.5 continued on next pge... 13
14 M. Ornels Mth 19 Lecture Notes Section 5.5 (continued) Evlute the definite integrl.. 3 dx 5x + 1 b. 1 xe x2 dx c. 2 (x 1)e (x 1)2 dx Integrls of Symmetric Functions Suppose f is continuous on [, ]. () (b) If f is even [f( x) = f(x)], then If f is odd [f( x) = f(x)], then f(x) dx = 2 f(x) dx =. f(x) dx. π/3 π/3 x 4 sin x dx 14
x = b a n x 2 e x dx. cdx = c(b a), where c is any constant. a b
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