MATH 1271 Monday, 21 November 218 Today: Section 5.4 - Indefinite Integrals and the Theorem Homework: 5-17 odd, 21-45 odd, 51-63 odd, 67, 71 1/13
Def Total displacement is the integral of the velocity function. Def Total distance traveled is the integral of the absolute value of the velocity function. Keep in mind that velocity is a signed quantity; it has magnitude and direction. (More generally, velocity is a vector quantity.) 2/13
Ex. Suppose that a particle moves along a line, to the right when v(t) > and to the left when v(t) <, with velocity v(t) = cos(t), in meters per second, from t =tot =2fi seconds. Find the total displacement and the total distance traveled of the particle on that time interval. For total displacement, we ll look at v(t) = cos(t): cos(t) total displacement = 2fi cos(t) dt 2fi =sin(t) - =sin(2fi) sin() = meters 3/13
Ex. Suppose that a particle moves along a line, to the right when v(t) > and to the left when v(t) <, with velocity v(t) = cos(t), in meters per second, from t =tot =2fi seconds. Find the total displacement and the total distance traveled of the particle on that time interval. Now, for total distance traveled, we want to look at cos(t) : cos(t) total distance = = 2fi fi/2 cos(t) dt cos(t) dt + 3fi/2 fi/2 2fi ( cos(t)) dt + cos(t) dt 3fi/2 4/13
Ex. Suppose that a particle moves along a line, to the right when v(t) > and to the left when v(t) <, with velocity v(t) = cos(t), in meters per second, from t =tot =2fi seconds. Find the total displacement and the total distance traveled of the particle on that time interval. total distance = = 2fi fi/2 =sin(t) - cos(t) dt cos(t) dt + fi/2 3fi/2 fi/2 +( sin(t)) - 2fi ( cos(t)) dt + 3fi/2 fi/2 +sin(t) - =(1 ) + (1 ( 1)) + ( ( 1)) =1+2+1 = 4 meters 2fi 3fi/2 3fi/2 cos(t) dt 5/13
Ex. A particle moving along a line, to the right when v(t) > and to the left when v(t) <. A graph of its velocity v(t) in feet per second is shown below. What is the total displacement of the particle from its initial position after 5 seconds? (2, 5) (5, 5) (4, 5) (a) feet (b) 5feet (c) 1 feet (d) 12.5 feet 6/13
Ex. A particle moving along a line, to the right when v(t) > and to the left when v(t) <. A graph of its velocity v(t) in feet per second is shown below. What is the total displacement of the particle from its initial position after 5 seconds? (2, 5) (5, 5) (4, 5) (a) feet (b) 5feet (c) 1 feet (d) 12.5 feet 7/13
Ex. A particle moving along a line, to the right when v(t) > and to the left when v(t) <. A graph of its velocity v(t) in feet per second is shown below. What is the total displacement of the particle from its initial position after 5 seconds? (2, 5) (5, 5) (4, 5) (a) feet (b) 5feet (c) 1 feet (d) 12.5 feet 8/13
Indefinite Integrals When we write an integral sign without endpoints of the interval of integration, this is our notation for the general antiderivative of the integrand function,and it is sometimes called the indefinite integral of the function. The definite integral is a NUMBER. The indefinite integral is a FAMILY OF FUNCTIONS. 9/13
Indefinite Integrals The definite integral is a NUMBER. fi fi fi sin(x) dx =( cos(x)) - fi =( cos(fi)) ( cos( fi)) =1 1 = The indefinite integral is a FAMILY OF FUNCTIONS. sin(x) dx = cos(x)+c 1 / 13
Indefinite Integrals The definite integral is a NUMBER. fi fi fi sin(x) dx =( cos(x)) - fi =( cos(fi)) ( cos( fi)) =1 1 = Ω number The indefinite integral is a FAMILY OF FUNCTIONS. sin(x) dx = cos(x)+c Ω family of function 11 / 13
Indefinite Integrals 1 Ô Ex. Find x 3 + 3Ô 2 x 2 dx. 1 Ô x 3 + 3Ô 2 x 2 dx = 1 2 x 3 2 2 + x 3 dx = 1 3 3 2 +1x = 1 5 2 2 +1 + 1 x 5 1 2 + x 5 3 + C 5 3 = 2 5 x 5 2 + 3 5 x 5 3 + C 2 2 3 +1x 3 +1 + C 12 / 13
Applications If an object moves along a straight line with position function s(t), then its velocity is v(t) =s Õ (t), so t2 v(t) dt = s(t 2 ) s(t 1 ). t 1 Use this to calculate displacement and distance! The acceleration of the object is a(t) =v Õ (t), so t2 t 1 a(t) dt = v(t 2 ) v(t 1 ) is the change in velocity from time t 1 to time t 2. Volume of water, concentration of product, mass of rod, etc. (see page 46) 13 / 13