This week we started a new chapter. We started with a chapter 6.2 and it was very similar to what we learned in fall. We basically used u-substitutions on definite integrals.
1) Decide what to choose for U.
2) calculate du/dx
3) Using the relationship between du and dx, change the original definite integrals in terms of x to in terms of u.
4) Calculate the simplified definite integrals (in u).
5) Replace u with the original terms.
But, I had hard time picking appropriate terms for u. Mostly, Us are something inside the parentheses or under the roots. But, when the equation includes trigonometry terms, it is harder to pick something for u. While doing homework and CCC, I kind of saw the pattern. Most questions give some relevant trigonometry terms, such as tan(x) and {sec(x)}^2.
On the homework, the most difficult question was #47.
∫ {sin(2x)}^3 dx Hint: (sin 2x)^2=1-(cos 2x)^2
The explanation is at the bottom.
On chapter 6.1, I learned definite and infinite integrals and slope fields. Slope fields are like the easiest part in calculus so far. (I am not sure about next week or applications.) We are given the equation for the derivative and we plug in x-coordinates and y-coordinates to the derivative, and that gives a slope at each point.
It was interesting that the slope fields of cosine function are also odd and the slope fields of sine function are even, and that both of them are oscillating.
Overall, my participation was good. I asked lots of math questions to my group and they are smart! So, CCC was really helpful.
1) Decide what to choose for U.
2) calculate du/dx
3) Using the relationship between du and dx, change the original definite integrals in terms of x to in terms of u.
4) Calculate the simplified definite integrals (in u).
5) Replace u with the original terms.
But, I had hard time picking appropriate terms for u. Mostly, Us are something inside the parentheses or under the roots. But, when the equation includes trigonometry terms, it is harder to pick something for u. While doing homework and CCC, I kind of saw the pattern. Most questions give some relevant trigonometry terms, such as tan(x) and {sec(x)}^2.
On the homework, the most difficult question was #47.
∫ {sin(2x)}^3 dx Hint: (sin 2x)^2=1-(cos 2x)^2
The explanation is at the bottom.
On chapter 6.1, I learned definite and infinite integrals and slope fields. Slope fields are like the easiest part in calculus so far. (I am not sure about next week or applications.) We are given the equation for the derivative and we plug in x-coordinates and y-coordinates to the derivative, and that gives a slope at each point.
It was interesting that the slope fields of cosine function are also odd and the slope fields of sine function are even, and that both of them are oscillating.
Overall, my participation was good. I asked lots of math questions to my group and they are smart! So, CCC was really helpful.