Announcements Topics: - sections 7.1 (differential equations), 7.2 (antiderivatives), and 7.3 (the definite integral +area) * Read these sections and study solved examples in your textbook! Homework: - review lecture notes thoroughly - work on practice problems from the textbook and assignments from the coursepack as assigned on the course web page (under the SCHEDULE + HOMEWORK link)
Solving Pure-Time DEs The general form of a pure-time differential equation is df dx = f (x) where F(x) is the unknown state variable and is the measured rate of change. f (x) Examples: (a) df dx = 4x 3 +1 (b) dy dx = 5ex + 1 1+ x 2
Solving Pure-Time DEs Solve each by guess and check. Ask yourself: What function has this as its derivative? Examples: df (a) (b) dx = 4x 3 +1 dy dx = 5ex + 1 1+ x 2
Antiderivatives/Indefinite Integrals An antiderivative (or indefinite integral) of a function f (x) is a function F(x) with derivative equal to f (x). F'(x) = f (x) F(x) = f (x)dx An antiderivative F(x) is a solution to the puretime differential equation df dx = f (x).
Initial Value Problems A differential equation has a whole family of solutions. Example: y'= 2x 4 y = x 2 4x + C 10 5-2.5 0 2.5 5 7.5 10-5
Initial Value Problems An initial value problem provides an initial condition so you can find a particular solution. Example: y'= 2x 4, y(0) = 3 y = x 2 4x + 3 10 5-2.5 0 2.5 5 7.5 10-5
Antiderivatives/Indefinite Integrals Theorem 7.2.1: If F(x) is an antiderivative of f(x), then the most general antiderivative of f(x) is F(x)+C; i.e., f (x)dx = F(x) + C where C is a real number. If an initial value of the solution is given, then we can solve for C to find a specific or particular antiderivative of f(x).
Rules for Antiderivatives The Power Rule for Integrals x n dx = x n +1 n +1 + C for n 1 Example: Integrate each. (a) x 7 dx (b) (c) 1 t dt 4 xdx
Rules for Antiderivatives The Constant Multiple Rule for Integrals Suppose f (x)dx = F(x) + C. Then for any constant af (x)dx = af(x) + C'. a. The Sum Rule for Integrals Suppose f (x)dx = F(x) + C and g(x)dx = G(x) + C'. Then [ f (x) + g(x)]dx = f (x)dx + g(x)dx = F(x) + G(x) + C''.
Summary Of Some Basic Integration Formulas x n dx = x n +1 n +1 + C for n 1 cos xdx = sin x + C x 1 dx = 1 x dx = ln x + C sin xdx = cos x + C e x dx = e x + C sec 2 xdx = tan x + C 1 dx = arctan x + C 2 1+ x
Some More Examples Example 1: Integrate. (a) (5x 4 3 x )dx (b) (sec 2 x + e 4 x )dx Example 2: Solve the differential equation with initial condition f '(x) = 2 x + 1 2 x f (0) = 0.
Even More Examples Example 1: Integrate. (a) Kl(γ +1) 2 d 4 dd (b) Kl(γ +1) 2 d 4 dγ Example 2: True or False? Explain. (a) 1 arctan x dx = (b) e x 1+ x 2 2 dx = e x2 2x + C
Summary Of Some Basic Integration Formulas x n dx = x n +1 n +1 + C for n 1 cos xdx = sin x + C x 1 dx = 1 x dx = ln x + C sin xdx = cos x + C e x dx = e x + C sec 2 xdx = tan x + C 1 dx = arctan x + C 2 1+ x
Area How do we calculate the area of some irregular shape? For example, how do we calculate the area under the graph of f on [a,b]? Area =?
Area Approach: We approximate the area using rectangles. number of rectangles: width of each rectangle: n = 4 Δx = b a n x 0 = x 1 x 2 x 3 = x 4
Area Left-hand estimate: Let the height of each rectangle be given by the value of the function at the left endpoint of the interval. x 0 = x 1 x 2 x 3 = x 4
Area Left-hand estimate: Area f (x 0 )Δx + f (x 1 )Δx + f (x 2 )Δx + f (x 3 )Δx ( f (x 0 ) + f (x 1 ) + f (x 2 ) + f (x 3 ))Δx 3 i= 0 f (x i )Δx Riemann Sum
Area Right-hand estimate: Let the height of each rectangle be given by the value of the function at the right endpoint of the interval. x 0 = x 1 x 2 x 3 = x 4
Right-hand estimate: Area Area f (x 1 )Δx + f (x 2 )Δx + f (x 3 )Δx + f (x 4 )Δx ( f (x 1 ) + f (x 2 ) + f (x 3 ) + f (x 4 ))Δx 4 i=1 f (x i )Δx Riemann Sum
Area Midpoint estimate: Let the height of each rectangle be given by the value of the function at the midpoint of the interval. x 1 x 2 x 3 x 4
Area Midpoint estimate: Area f (x 1 * )Δx + f (x 2 * )Δx + f (x 3 * )Δx + f (x 4 * )Δx ( f (x 1 * ) + f (x 2 * ) + f (x 3 * ) + f (x 4 * ))Δx 4 i=1 f (x i * )Δx Riemann Sum
Area How can we improve our estimation? Increase the number of rectangles!!! Area 16 i=1 f (x i * )Δx How do we make it exact? Let the number of rectangles approach infinity!!!
Riemann Sums and the Definite Integral Definition: The definite integral of a function f on the interval from a to b is defined as a limit of the Riemann sum b a f (x)dx = lim n n i=1 f (x i * )Δx where [x i 1, x i ] is some sample point in the interval and x i * Δx = b a n.
The Definite Integral Interpretation: If f 0, then the definite integral is the area under the curve y = f (x) from a to b. Area = b a f (x)dx
The Definite Integral Example: Estimate the value of 1 e t2 dt 0 rectangles and left-endpoints. using two 1.25 1 0.75 0.5 0.25-1 -0.75-0.5-0.25 0 0.25 0.5 0.75 1 1.25
The Definite Integral Example: Estimate the value of 1 e t2 dt using two 0 rectangles and right-endpoints. 1.25 1 0.75 0.5 0.25-1 -0.75-0.5-0.25 0 0.25 0.5 0.75 1 1.25
The Definite Integral Example: Estimate the value of 1 0 rectangles and midpoints. e t2 dt using two 1.25 1 0.75 0.5 0.25-1 -0.75-0.5-0.25 0 0.25 0.5 0.75 1 1.25
The Definite Integral Interpretation: If f is both positive and negative, then the definite integral represents the NET or SIGNED area, i.e. the area above the x-axis and below the graph of f minus the area below the x-axis and above f 4 1 f (x)dx = net area
Definite Integrals and Area Example: Evaluate the following integrals by interpreting each in terms of area. (a) 1 0 1 x 2 dx (b) 3 0 (x 1) dx (c) π π sin x dx
Properties of Integrals Assume that f(x) and g(x) are continuous functions and a, b, and c are real numbers such that a<b. a a (1) f (x) dx = 0 b a (2) f (x) dx = a b f (x) dx b (3) c f (x) dx = c f (x) dx a b a
Properties of Integrals Assume that f(x) and g(x) are continuous functions and a, b, and c are real numbers such that a<b. b a (4) ( f (x)± g(x) )dx = f (x) dx ± g(x) dx b a (5) c dx = c(b a) b a b a
Properties of Integrals (6) Suppose f(x) is continuous on the interval from a to b and that a c b. c a f (x) dx b c f (x) dx Then b f (x) dx = f (x) dx + f (x) dx. a c b a c
Properties of Integrals (7) Suppose f(x) is continuous on the interval from a to b and that m f (x) M. Then m(b a) b a f (x) dx M(b a).