2. Sect. 4.1 Antiderivatives Sect. 4.2 Area Sect. 4.3 Riemann Sums/Definite Integrals Sect. 4.4 FTC and Average Value Sect. 4.5 Integration by Substitution Sect. 4.6 Numerical Integration and Trapezoidal Rule Particle Motion CHAPTER FOUR: INTEGRATION

3. Write the general solution of a differential equation. Use indefinite integral notation for antiderivatives. Use basic integration rules to find antiderivatives. Find a particular solution of a differential equation. Sect. 4.1 Antiderivatives and Indefinite Integration

4. A physicist who knows the velocity of a particle might wish to know its position at a given time. A biologist who knows the rate at which a bacteria population is increasing might want to deduce what the size of the population will be at some future time. Introduction

5. DIFFERENTIAL EQUATIONS A differential equation is an equation that contains a derivative. By antidifferentiating, we can solve this equation for y.

6. Integrand Variable of Integration Constant of Integration Read: The antiderivative of f with respect to x or the indefinite integral of f with respect to x is equal to….. Indefinite Integrals The family of all functions that are antiderivatives of f (x) is called the indefinite integral. Integral Sign

7. Some General Rules They are just the derivative rules in reverse. Differentiation Formula Integration Formula

8. Some More General Rules Differentiation Formula Integration Formula Sum / Difference Rule for Integrals Power Rule for Integrals

9. Integrate

10. OTHER INTEGRALS THAT SHOULD BE KNOWN

11. Integrate

12. You Try… Integrate the following.

13. Rewriting before integrating

14. You Try… Find the antiderivative.

15. Initial Conditions and Particular Solutions When we find antiderivatives, we are creating a family of curves for each possible value of c, called the general solution. C=0 C=1 C=3 When we find the value of C, given a point on y (called an initial condition), we are able to find a particular solution.

16. Ex: Find the general solution of the equation F’(x) = and find the particular solution given the point F(1) = 0. Now plug in (1,0) and solve for C. 0 = -1 + C C = 1

17. Find the function f (x) for which and f (1) = 3. You Try… So, 3 = 1 + C and therefore, C = 2. Therefore, our function is

18. Given: Integrate: Substitute x=0 and 6 for f’(0) Thus c = 6 Integrate again: Substitute x = 0 and f(0)=3 c= 3 Find f(x) Answer: You Try…

19. Closure Explain how to integrate Explain how to check your answer the problem above.

20. SUMMARY for Integration A differential equation is an equation that contains a derivative. You can solve them analytically using integration. The integral of a derivative is called the antiderivative. Don’t forget to include an arbitrary constant. There are infinite solutions to a differential equation, but with an initial value you can solve for an arbitrary constant to help you calculate the final equation (particular solution). You SHOULD memorize the table of integrals on pgs. 244 & 391!! In addition, memorizing derivatives are very, very important!