The first half of the week we focused on 5.2 Definite Integrals, while the second half of the week we focused on 5.3 Definite Integrals and Anti-Derivatives. For section 5.2 we were reminded of sigma notation, and learned the definite integral of a continuous function which would be the sum of f of C sub K times the change in X sub K as the limit approaches infinity. We looked at how to find the area under a curve if f(x) is nonnegative or if f(x) is nonpositive. It's important you use nonnegative and nonpositive instead of negative or positive because 0 could be included in the interval. Net area is found by taking the area above the curve then subtracting the area below the curve.
Section 5.3 was about the rules of definite integrals. The 6 rules were used in solving different problems dealing with definite integrals. We also learned the Mean Value Theorem for Definite Integrals. The theorem says that if f(x) is continuous on [a,b] then at some point c in [a,b] and f(c)=1/(b-a) * the integral f(x)dx on the interval [a,b]. An example of this would be to find the average value of f(x) if f(x)=4-x^2, [0,3]. I'd write the integral of (4-x^2)dx on the interval 0 to 3 and find that it equals 3. Then I'd plug the numbers into the formula to get f(c)=1/(3-0) * 3 to get the final answer to be 1.
Section 5.3 was about the rules of definite integrals. The 6 rules were used in solving different problems dealing with definite integrals. We also learned the Mean Value Theorem for Definite Integrals. The theorem says that if f(x) is continuous on [a,b] then at some point c in [a,b] and f(c)=1/(b-a) * the integral f(x)dx on the interval [a,b]. An example of this would be to find the average value of f(x) if f(x)=4-x^2, [0,3]. I'd write the integral of (4-x^2)dx on the interval 0 to 3 and find that it equals 3. Then I'd plug the numbers into the formula to get f(c)=1/(3-0) * 3 to get the final answer to be 1.