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4.1 Antiderivatives and Indefinite Integration What if you were asked to find a function whose derivative is 4x3? In other words, we are going to work the derivative backwards. What we are trying to find is called an _______________________ of the function. If we call f(x) = 4x3, what are possible solutions for F(x) such that F’(x) = f(x) ? If we try to think backwards, possible solutions are: ____________ ___________ General Solution:___________________________ Notations for antiderivatives: The process of finding all solutions of this equation is called ___________________________ or ___________________________ The general solution is denoted by y f ( x)dx _________ where C is the constant of integration. The expression above is read as “the ________________ or _____________ of f with respect to x”. Differentiation and Integration are ________________ of each other. Basic Integration Rules 0dx _________ kdx _________ kf ( x)dx k f ( x)dx [ f ( x) g ( x)] f ( x)dx g ( x)dx x dx _____________ cos xdx ____________ sin xdx ____________ sec xdx ____________ sec x tan xdx ____________ n 2 csc 2 xdx _______________ csc x cot xdx _____________ Examples: Find the general solution of the differential equation. 1) dy 10 x 4 dx Examples: Evaluate the indefinite integral. 2) (x 4) 4 3 4)dx x3 dx 3) (1 3t )t dt 5) (x 3 3 3cos x)dx 6) x 1 dx x 7) sec y tan y sec y dy Particular Solutions: Find ____ Examples: 8) Find the particular solution of dy 2( x 1) , given that the point (3,2) is on the curve. dx 9) Solve the differential equation. f ''( x) x 2 , f '(0) 6, f (0) 3 4.1 Part 2 Particle Motion In questions about particle motion, the particle may be moving in one dimension along a line, usually the x-axis (though it could also be moving along the y-axis). The particle may also be moving in two dimensions, in which case it may be referred to as a freely falling object. The ______________ of the particle at time t, usually denoted x(t) or s(t), is its location on the number line. Position is expressed in linear units such as meters or feet. The ______________ of the particle is the rate of change of position with respect to time, or the derivative of position. Velocity tells us how fast and in what direction the particle is moving. Velocity is expressed as a ratio of linear units and time units, such as m/sec or ft/sec. The ______________ of the particle tells us only how fast the particle is moving. Speed is the absolute value of velocity. The ______________ of the particle is the rate of change of velocity with respect to time, or the derivative of velocity. Acceleration is expressed as a ratio of linear units and time units squared, such as m/sec2 or ft/sec2. The _________________________ by the particle is the integral of the absolute value of the velocity equation. Particle Velocity Particle at rest Particle moves to right (or up) Particle moves to left (or down) Particle changes direction Example 1 Suppose a particle is moving along a line with its position at time t given by for t 0 (t is in seconds and s is in feet). (a) Find the velocity function. (b) Find v(0) and v(2). s(t ) t 3 3t 2 , (c) When does the particle change direction? Where is the particle when it changes direction? (d) Write a description of the motion of the particle from t = 0 to t = 2. (e) Find the distance traveled by the particle in the first two seconds. Example 2 A particle moves along the x-axis so that its velocity at time At time t is given by v(t ) 2 cos t . t 0 , the particle is at position x 1 . (a) Find the acceleration of the particle at time (b) Find all times t in the open interval Justify your answer. t 2. 0 t 3 when the particle changes direction. (c) Find an equation for the position of the particle. (d) Find the total distance traveled by the particle from time t 0 to time t 3 . 4.3 Riemann Sums and Definite Integrals Definition of a Riemann Sum Let f be defined on the closed interval [a, b] and let be a partition of [a, b] given by a = x0 < x1 < x2 < . . . < xn - 1 < xn = b, where xi is the length of the ith subinterval. If ci is any point in the ith subinterval, then the sum n f ( c ) x , i 1 i i x i - 1 c i xi is called a Riemann sum of f for the partition . Plain English: You are breaking an area up into rectangles of equal width, calculating the areas of these rectangles, and adding their areas Examples: 1) Consider the graph at right. Approximate the Riemann sum, using the midpoints of five subintervals of equal length. 2) Consider the graph of f ( x) 5 on the x 1 2 interval [0,3]. Approximate the Riemann sum, using the right endpoints of six subintervals of equal length. Definition of Definite Integral (Informal) As x approaches 0 in the Riemann sum formula above (that is, as the rectangles become skinnier and skinnier), the Riemann sum becomes the Definite Integral (the true area of the region). The Definite Integral as the Area of a Region If f is continuous and nonnegative on the closed interval [a, b], then the area of the region bounded by the graph of f, the x-axis, and the vertical lines x = a and x = b is given by b A f ( x)dx a Examples: Find the definite integral. Draw a picture of the region described. 4 5 3dx 3) 4) ( x 3)dx 0 0 5 5) 1 (5 x )dx 3xdx 6) 5 2 Properties of Definite Integrals a 1. If f is defined at x = a, then f ( x)dx 0 . a a b b a f ( x)dx f ( x)dx . 2. If f is integrable on [a, b], then 3. If f is integrable on the three closed intervals determined by a, b, and c, then b c b a c f ( x)dx f ( x)dx f ( x )dx a 4. If a function f is continuous on the closed interval [a, b], then f is integrable on [a, b]. 3 Example 7: Given f ( x)dx 4 0 6 and f ( x)dx 1 ,find: 3 3 a) f ( x)dx 3 3 b) f ( x)dx 6 6 c) f ( x)dx 0 4.6 Numerical Integration – Trapezoidal Rule Example: Find the area under the curve using four trapezoids. Theorem: The Trapezoidal Rule Let f be continuous on [a,b]. b The Trapezoidal Rule for approximating f ( x)dx is given by a ba f ( x)dx f x0 2 f x1 ... 2 f xn 1 f xn 2n /2 cos xdx Example 1: Use the Trapezoidal Rule to approximate 0 for n = 4. Example 2: A table of values for a continuous function f is shown below. If four equal 2 subintervals of [0, 2] are used, what is the trapezoidal approximation of f ( x)dx ? 0 x f(x) 0 3 0.5 3 1.0 5 1.5 8 2.0 13 4.4 Part 1 The Fundamental Theorem of Calculus The Fundamental Theorem of Calculus If a function f is continuous on the closed interval [a,b] and F is an antiderivative of f on the interval [a,b], then b f ( x)dx F (b) F (a) a Examples: Evaluate the definite integral. 1 5 (3v 4)dv 1) 2) 2 2 /2 2t cos t dt 3) 1 du 2 u u 9 4) / 2 3 x xdx 0 2 5) 2 x 1 dx 6) Determine the area of the indicated region. 0 y 1 x2 4.4 Part 2 Average Value / Second Fundamental Theorem of Calculus Finding a definite integral on your calculator: 5 1) (3v 4) 2 2) The position function is x(t ) t 3 6t 2 9t 2,0 t 5 . Find the total distance the particle travels in 5 units of time. Definition of the Average Value of a Function on an Interval If f is integrable on the closed interval [a,b], then the AVERAGE VALUE of f on the interval is b 1 f ( x)dx b a a EXAMPLES: Find the average value of the function over the interval. 3) f(x) = x2 + 1 x2 [-2, 2] 4) The volume V in liters of air in the lungs during a 5-second respiratory cycle is approximated by the model V = 0.1729t + 0.1522t2 - 0.0274t3 where t is the time in seconds. Approximate the average volume of air in the lungs during one cycle. x Example: Let F ( x) cos tdt . Find F '( x ) . 0 The Second Fundamental Theorem of Calculus If f is continuous on an open interval I containing a, then, for every x in the interval, x d f (t )dt f ( x) dx a a = _______________ x = _______________ x Example 5: Find F '( x ) . t2 F ( x) 2 dt t 1 1 If the upper limit is not “just x” , use the chain rule to finish the problem. x3 Example 6: Find F '( x ) . F ( x) 1 x 3 2 dx More on the Second Fundamental Theorem of Calculus The graph of f shown consists of a semicircle and two line segments on the interval [-2,6]. x If g ( x) f (t )dt , use the graph of f to do the following: 2 (a) Find g (2), g (0), g (4), g (6) . (b) Find g '(2), g '(0), g '(2), g '(4), g '(6) (c) Find the intervals where g is decreasing. (d) Find the intervals where g is concave up. (e) Find the absolute extrema of g . (f) Find the x-coordinates of each point of inflection of g . x The graph of g (t ) is shown on the interval [0,6] and f ( x) g (t )dt . 0 (a) Find g (0), g (2) . (b) Find g '(0), g '(2) . (c) Find the critical numbers of f ( x ) , and identify each as a relative maximum, a relative minimum, or neither. (d) On what interval(s) is f ( x ) concave up and concave down? (e) On what interval(s) is f ( x ) increasing and/or decreasing? 4.5 Part 1 Integration by Substitution Antidifferentiation of a Composite Function Let g be a function whose range is an interval I, and let f be a function that is continuous on I. If g is differentiable on its domain and F is an antiderivative of on I, then f ( g ( x)) g '( x)dx F ( g ( x)) C If u = g(x), then du = g’(x)dx and f (u )du F (u ) C . Plain English: If, in differentiation, you would have to apply the chain rule to the function you are dealing with, you must make a u-substitution, where u is the function in question. You then find du and replace it in the integral set-up. Examples: Find the integral. 1) 2 4 2 x( x 1) dx 3) 4x (x 5) 5cos 5xdx 3 4 2)3 dx 1 2) 2 3 3x ( x 1) 2 dx 4) tan 5 x sec2 xdx Note: If you are missing a constant from the integrand when finding du, multiply and divide by the missing constant. Example: Evaluate the integral: 6) 8) 2 2 x( x 1) dx 5x x 2 3dx 7) 2 3x 2 x 5 dx 9) Solve. 4 dy dx x4 x2 8x 1 4.5 Part 2 Integration by Substitution If du is not found in the integrand, you will have to solve your du equation for dx, and then use substitution techniques. Sometimes you might even have to solve for x. Examples: Evaluate the integral. 1) x 2 x 1dx 2) x dx x2