Lecture 18 More on the Fundamental Theorem of Calculus

The Area function: For a function f and a fixed a is the “area under the curve from a to x xba Choosing “b” rather than “a” only changes the area function by a constant.

Fundamental Theorem of Calculus If f is continuous then = f(x) If f(x) is continuous on an interval I then the area function for f(x) is an antiderivative for f on I =

= = =

The green area is very close to the rectangle so its area is approximately = area of green-shaded figure = (approx) The continuity hypothesis says this gets better and better as h -> 0

= = = So =

Example: = =

Water is draining from a tank. At time t hrs the flow out of the tank is gallons per hour. How many gallons flow out of the tank during the second hour? Let V(t) = amount in the tank at time t. We want V(2)-V(1) V(2) – V(1) = 1 2 = ( ) () Ans = gallons

A car travels along the x-axis. At time t its velocity is given by V(t) = feet per second At time t = 3 seconds the car is at x = 25 feet. Where is the car at time t seconds? Let S(t) be the position). Since S is an antiderivative for V we have = S(3) = so

The parabola is the graph of The line is the graph of y = 1-x What is the area of the green region? b a Area under the parabola from a to b - area under the line from a to b Solve From quadratic equation x = (= a), x= (= b) = =