Chapter 4 Integration

SECTION 4.1 Antiderivatives and Indefinite Integration Calculus: Chapter 4 Section 4.1

Antiderivative A function F is an antiderivative of f on an interval I if F '( x) f ( x) for all x on I. Calculus: Chapter 4 Section 4.1

Representation of Antiderivatives If F is an antiderivative of f on I, then G is an antiderivative of f on the I if and only if G is of the form G( x) F( x) C for all x on I, where C is a constant. Calculus: Chapter 4 Section 4.1

Some Integral Words Differential Equation An equation with a derivative or differential term in it. General Solution The general solution of a differential equation is the antiderivative of the differential equation added to a constant Constant of Integration that s the C Calculus: Chapter 4 Section 4.1

Integral Notation If dy dx f( x) then dy f ( x) dx If we solve for y we are going to take the antiderivative of both sides of the differential equation... variable of integration y f ( x) dx F( x) C integrand constant of integration Calculus: Chapter 4 Section 4.1

Basic Integration Rules F '( x) dx F( x) C d dx f ( x) dx f ( x) Calculus: Chapter 4 Section 4.1

Basic Integration Rules Constant Rule: k dx kx C Sum/Difference: f ( x) g( x) dx f ( x) dx g( x) dx Power Rule: 1 n 1 n n1 x dx x C Calculus: Chapter 4 Section 4.1

Trig Functions sin x dx cos x C cos x dx sin x C sec x tan x dx sec x C 2 sec x dx tan x C csc xcot x dx csc x C 2 csc x dx cot x C Calculus: Chapter 4 Section 4.1

83.At the instant the traffic light turns green, a car that has been waiting at an intersection starts with a constant acceleration of 6 feet per second per second. At the same instant, a truck traveling with a constant velocity of 30 feet per second passes the car. (a)how far beyond its starting point will the car pass the truck? (b) How fast will the car be traveling when it passes the truck? Calculus: Chapter 4 Section 4.1

Calculus: Chapter 4 Section 4.2 SECTION 4.2 Area

Sigma Notation The sum of the first n terms a 1, a 2, a 3, a n is written as n i1 a a a a... a i 1 2 3 Where i is the index of summation, a i is the ith term of the sum, and the upper and lower bounds of summation are n and 1. n Calculus: Chapter 4 Section 4.2

Sigma Properties 1. n n k a k a i i1 i1 i 2. n n n a b a b i i i i i1 i1 i1 Calculus: Chapter 4 Section 4.2

Sigma Formulas 1. n i1 c c n 2. n i1 i nn ( 1) 2 Calculus: Chapter 4 Section 4.2

Sigma Formulas 3. n i1 i 2 n( n 1)(2n 1) 6 4. n i1 i 3 n 2 2 ( n1) 4 Calculus: Chapter 4 Section 4.2

Area Under the Curve Calculus: Chapter 4 Section 4.2

Finding the Area Using n Rectangles 1. Determine the expression for Dx 2. Determine the expression for x i 3. Determine the expression for f(x i ) 4. Approximate the area between the curve and the x- axis using the following summation: n i1 f ( x ) i Dx Calculus: Chapter 4 Section 4.2

Upper and Lower Sums Upper Sums are sums found using circumscribed rectangles; that is, the rectangles are above the curve Lower Sums are sums found using inscribed rectangles; that is, the rectangles are under the curve Lower and Upper are NOT to be confused with Left and Right Hand Sums. Calculus: Chapter 4 Section 4.2

Limits of Upper and Lower Sums Let f be continuous and non-negative on the interval [a, b]. The limits as n approaches infinity of both the lower and upper sums exist and are equal to each other. That is, n lim s( n) lim f ( m ) Dx n n i 1 n lim S( n) lim f ( M ) Dx n n i 1 i i Calculus: Chapter 4 Section 4.2

Definition of the Area of a Region in a Plane Let f be continuous and non-negative on the interval [a, b]. The area of the region bounded by the graph of f, the x-axis, and the vertical lines x = a and x = b is: n Area lim f ( c ) Dx n i 1 i x c x i 1 i i Calculus: Chapter 4 Section 4.2

The limit of the right-hand sum 2 3n 2 3n 4 3n 6 3n 2n lim... n n n n n n 2 2 2 2 Calculus: Chapter 4 Section 4.2

SECTION 4.3 Riemann Sums & The Definite Integral Calculus: Chapter 4 Section 4.3

Riemann Sums Let f be defined on the closed interval [a, b] and let D be a partition of [a, b] given by a x x x... x x b 0 1 2 n1 n Where Dx i is the width of the ith subinterval. If c i is any point in the ith subinterval, then the sum Is called a Riemann Sum of f for the partition D. n i1 f ( c ) i Dx i Calculus: Chapter 4 Section 4.3

Definite Integral If f is defined on the closed interval [a, b] and the limit n lim f ( c ) Dx i i D0 i 1 Exists, then f is integrable on [a, b] and the limit is denoted by: n lim f ( c ) x f ( x) dx b D i i D0 a i 1 Calculus: Chapter 4 Section 4.3

Continuity and Integrability If f is continuous on the closed interval [a, b], then f is integrable on that interval. Calculus: Chapter 4 Section 4.3

The Difference Between AREA and DEFINITE INTEGRAL Area is always a positive quantity Area = b a f ( x) dx A definite integral can be negative, and anything below the x-axis will have a negative area Calculus: Chapter 4 Section 4.3

Properties of Integrals a 1. f x dx a ( ) 0 2. b a f ( x) dx f ( x) dx b a 3. b c b f ( x) dx f ( x) dx f ( x) dx a a c Given that c is on [a, b] Calculus: Chapter 4 Section 4.3

Properties of Integrals b 4. a k f ( x) dx k f ( x) dx b a 5. b b b f ( x) g( x) dx f ( x) dx g( x) dx a a a Calculus: Chapter 4 Section 4.3

Calculus: Chapter 4 Section 4.3

SECTION 4.4 The Fundamental Theorem of Calculus Calculus: Chapter 4 Section 4.4

The Fundamental Theorem of Calculus 1 st Form If a function f is continuous on the closed interval [a, b], and F is the antiderivative of f on the interval [a, b], then: b f ( x ) dx F ( b ) F ( a ) a Calculus: Chapter 4 Section 4.4

Mean Value Theorem for Integrals If a function f is continuous on the closed interval [a, b], then there exists a number c on the interval [a, b] such that b f ( x ) dx f ( c ) ( b a ) a Calculus: Chapter 4 Section 4.4

Average Value of a Function If a function f is integrable on [a, b], then the average value of f on the interval is: 1 b ( ) b a f x dx a Calculus: Chapter 4 Section 4.4

AVERAGE RATE OF CHANGE of f(x) f ( b) f ( a) b AVERAGE VALUE of f(x) a b 1 a b a f ( x ) dx Calculus: Chapter 4 Section 4.4

Fundamental Theorem of Calculus 2 nd Form If f is continuous on an open interval I containing a, then, for every x in the interval, d dx a x f ( t) dt f ( x) Calculus: Chapter 4 Section 4.4

The rate of growth of the money in Jed s bank account, in dollars per month, is modeled by the function R(t). Find the AVERAGE rate of growth of the amount of money in Jed s bank account. Calculus: Chapter 4 Section 4.4

Calculus: Chapter 4 Section 4.4

Calculus: Chapter 4 Section 4.5 SECTION 4.5 Substitution

Substitution Guidelines 1. Choose a substitution u = g(x). Usually, it is best to choose the inner part of a composite function, such as a quantity raised to a power. 2. Compute du = g (x)dx. 3. Rewrite the integral in terms of u. 4. Find the resulting integral in terms of u. 5. Replace u by g(x) to obtain an antiderivative in terms of x. 6. If you have a definite integral, make sure you change the limits of integration to be in terms of u before you integrate. Calculus: Chapter 4 Section 4.5

Even & Odd Functions Let f be integrable on the closed interval [-a, a]: 1. EVEN FUNCTIONS a a f ( x) dx 2 f ( x) dx 0 a 2. ODD FUNCTIONS a a f ( x) dx 0 Calculus: Chapter 4 Section 4.5

Calculus: Chapter 4 Section 4.6 SECTION 4.6 Numerical Integration

Trapezoidal Rule Let f be a continuous function on [a, b]. The Trapezoidal b Rule for approximating f ( x) dx is a b a b a f ( x) dx f ( x0 ) 2 f ( x1 ) 2 f ( x2)... 2 f ( xn 1) f ( xn) 2n Calculus: Chapter 4 Section 4.6

Simpson s Rule Let f be a continuous function on [a, b]. Simpson s Rule b for approximating f ( x) dx is a b a f ( x 0) 4 f ( x 1) 2 f ( x 2) 4 f ( x 3)... 4 f ( x n1) f ( x n) 3n Calculus: Chapter 4 Section 4.6

Calculus: Chapter 4 Section 4.6