I. The Fundamental Theorem of Calculus Part I.

I. The Fundamental Theorem of Calculus Part I.

7.1 Integral as Net Change b I. Net Area f ( x)dx net area bounded by f(x) and the xa axis from a to b equals the NET CHANGE OF THE PRODUCT OF f(x)dx OVER [a, b]. 2) This is also the net change of the product f(x)*dx over [a,b] The units of the area under the curve is the product of the units of the two axes. The position of a particle whose velocity function is given by v(t) is t s (t ) s (0) v( x)dx 0 (starting position + how far you travelled).

II. Examples EX 1: A particle moving along a horizontal line 2 v ( t ) 6 t 18t 12 has velocity where t is given in minutes and s is given in feet. Find the particles position at t = 4 if s(0) = 5. 4 s (4) s (0) v(t )dt 0 4

s (4) 5 6t 18t 12 dt 2 0 s (4) 5 2t 9t 12t 3 2 4 0 s (4) 37 feet EX 2: Find the total distance travelled in example 1. 4 TDT v(t ) dt v(t ) 0

v(t ) 0 v(t ) 0 0 4 1 TDT 6t 2 18t 12 dt 2 0 2 6t 18t 12 0 6 t 2 t 1 0 t 2 or t 1 1

TDT v(t )dt 0 2 4 v(t )dt v(t )dt 1 TDT 34 feet 2 EX 3: The world use of copper is increasing at a rate give by f (t ) 15e.015t in millions of tons of copper from 2001 forward. (a) Find the total amount of copper used from the beginning of 2001 to the beginning of 2006 5 f (2006) 15e.015t dt

0 5 15 .015t f (2006) e 77.884 mil. of tons .015 0 (b) Find the total amount used from the beginning of 2006 to the end of 2008: 8 .015t 15e 5 dt 49.613 mil. of tons

EX 4: The density of cars (in cars per mile) on the highway approaching Disney World is f ( x) 200[4 ln(2 x 3)] where x = the distance in miles from the Disney entrance. Find the total number of cars on the 10 mile stretch. 10 200[4 ln(2 x 3)]dx 3118 cars 0 EX 5: In 2005, the worlds yearly petroleum consumption rate was about 22 billion barrels per year and was growing at 5%. How many years after 2005 is the worlds estimated oil reserves of 900 billion barrels likely to last? 45 .05 H e .05t

R(t ) 22e H 22 e.05t dt 900 0 H 22 20e 440 e 1 900 .05t .05 H 0

900 1 22 67 .05 H e 22 67 .05 H ln 22 H 22.273 years

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