Calculus AB Rate integration

Question

4. Football fans enter a sports stadium at a rate of r(t) people per hour, where hours If there are 18, 000 people in the stadium at the end of the first hour, write an expression that would represent the number of fans inside the stadium by the end of the second hour

Answer 1

The rate of fans arriving an hr can be represented by the slope of the line m = 18,000 fans/hr, the rate can also be written in slope equation form as m = y-yo/ x-xo, where xo and yo are the points at the origin ( xo,yo) with value (0,0). Therefore m = y-0/x-0, m = y/x , and y = m x. y = 18,000 fans/hr . x

at the end of one hr:

y = 18000 fans/hr ( 1hr) = 18,000 fans, and the end of 2 hrs, y = 18,000 fans/hr ( 2hrs) = 36,000 fans

Also the line y = 18,000. x is a straight line with an increasing slope and it represents the rate of change of the number of fans, so the area under the straight line can be represented by the sliver Δx multiplied by the height of the curve between the x axis and the line. If a finite number of Δx are added together between 0 and 2 hrs, where 0 and 2 are bounds of integration then the change in y can be written as:

Δy = 18,000 Δx , by integrating both sides and setting the bounds of integration between 0 and 2

_{2 2}

y = ∫₀ 18,000 dx = 18,000. x ] ₀ = 18,000 . 2 = 36,000 fans