1. Indefinite Integrals, Applications Section 6.1b

2. The set of all antiderivatives of a function is the indefinite integral of with respect to and is denoted by Integral Sign Integrand Variable of Integration Also, recall that a function is an antiderivative of if Definition: Indefinite Integral

3. Then all antiderivatives of a function vary by constants: What keeps this integral from being “definite”??? The constant C is the constant of integration and is an arbitrary constant. When we find we have integrated or evaluated the integral…

4. Integral Formulas Indefinite IntegralReversed Derivative Formula 1. (a) (b) 2.

5. Integral Formulas Indefinite IntegralReversed Derivative Formula 3. 4. 5.

6. Integral Formulas Indefinite IntegralReversed Derivative Formula 6. 7. 8.

7. Using Integral Formulas Evaluate:

8. Properties of Indefinite Integrals Let k be a real number. 1. Constant Multiple Rule: If k = –1, then: 2. Sum and Difference Rule:

9. Integrating Term by Term Evaluate But we can simply combine all of these constants!!!

10. Do Now – p.314, #55 How long did it take the hammer and feather to fall 4 ft on the moon? Solve the following initial value problem for s as a function of t. Then find the value of t that makes s equal to 0. Differential equation: Initial conditions: and when Velocity:

11. Do Now – p.314, #55 How long did it take the hammer and feather to fall 4 ft on the moon? Solve the following initial value problem for s as a function of t. Then find the value of t that makes s equal to 0. Differential equation: Initial conditions: and when Position:

12. Do Now – p.314, #55 How long did it take the hammer and feather to fall 4 ft on the moon? Solve the following initial value problem for s as a function of t. Then find the value of t that makes s equal to 0. Differential equation: Initial conditions: and when Solving, we have Take the positive solution… They took about 1.240 seconds to fall

13. More Application Problems A right circular cylindrical tank with radius 5 ft and height 16 ft that was initially full of water is being drained at the rate of 0.5 x ft /min (x = water’s depth). Find a formula for the depth and the amount of water in the tank at any time t. How long will it take the tank to empty? 3 x Diff Eq:

14. More Application Problems A right circular cylindrical tank with radius 5 ft and height 16 ft that was initially full of water is being drained at the rate of 0.5 x ft /min (x = water’s depth). Find a formula for the depth and the amount of water in the tank at any time t. How long will it take the tank to empty? 3 Initial Condition: Solve Analytically: Diff Eq:

15. More Application Problems A right circular cylindrical tank with radius 5 ft and height 16 ft that was initially full of water is being drained at the rate of 0.5 x ft /min (x = water’s depth). Find a formula for the depth and the amount of water in the tank at any time t. How long will it take the tank to empty? 3 Solve Analytically: Initial Condition:

16. More Application Problems A right circular cylindrical tank with radius 5 ft and height 16 ft that was initially full of water is being drained at the rate of 0.5 x ft /min (x = water’s depth). Find a formula for the depth and the amount of water in the tank at any time t. How long will it take the tank to empty? 3 Equation for volume: At what t is V = 0?minutes (The tank will be empty in about 21 hours)

17. More Application Problems You are driving along a highway at a steady 60 mph (88 ft/sec) when you see an accident ahead and slam on the brakes. What constant deceleration is required to stop your car in 242 feet? First, solve the following initial value problem: Differential Equation:(k constant) Initial Conditions: andwhen Velocity: Solution:

18. More Application Problems You are driving along a highway at a steady 60 mph (88 ft/sec) when you see an accident ahead and slam on the brakes. What constant deceleration is required to stop your car in 242 feet? Next, find the value of t that makes ds/dt = 0: Velocity:Solution:

19. More Application Problems You are driving along a highway at a steady 60 mph (88 ft/sec) when you see an accident ahead and slam on the brakes. What constant deceleration is required to stop your car in 242 feet? Velocity:Solution: Finally, find the value of k that makes s = 242 for the previously found value of t : You would need to decelerate at this constant rate in order to stop in 242 feet!!!