Please find the area bounded by the curves y^2+x=4 and y=x^3. (Please set up the integral and sketch the graph, labeling intersections. You may use NINT to compute.)


The first thing you should do is to solve for a variable. This case you use y. The solved equations are:

Then you would graph the two curves and find the intersects (the bounds).

To find the area of this area, you have to subtract the left curve from the right curve instead of the bottom curve from the top curve. The integral should look like this:

You then can type (4-y^2)-(cube root y),y,-2.307,1.188 into NINT to get the area under the curve.

Please find the area bounded by the curves f(x)=-cos(x) and g(x)=-x^2+5. (Please set up the integral. You may use NINT to compute)


The first thing you should do is graph both curves to get an idea of what the area looks like and where the curves are compared to each other. 

The less bold line is f(x) and the bolder line is g(x). As you see, g(x) is above f(x) and so the integral should be g(x)-f(x). To find the bounds, you find where the curves intersect and use the x-value. These values are -2.1 and 2.1. We have our bounds and we have what is in the integral, so let's put everything together. The final product should look like this:

After you find the integral, you type in all the information into NINT like so: (-x^2+5)-(-cos(x)),x,-2.1,2.1 and the calculator spits the answer.

Find the total distance traveled by the particle.
Given v(t)=4cos(t), 0 is less than or equal to t which is less than or equal to 2 pi.

To find the total distance traveled by the particle, you solve the integral of the given equation, but with a twist. You have to have the absolute value of that equation. Think about when you find if a particle is moving left or right. The positive part of the graph is the particle moving right, moving in a positive direction. The best way to solve the integral, use NINT. You should type (abs(4cosx),x,0,2 pi). Remember to close the parenthesis of the absolute value or you will get an error message. With these steps, you are sure to get the right answer.

Let's do another!

Given v(t)=.25t^3-t, 0 is less than or equal to t which is less than or equal to 3.


Follow the same steps as above.

To find a particles displacement, you take the integral of the given equation. Doing this by hand and then by using NINT is a good way to make sure you have the correct answer. Your anti-derivative for this equation should be -4sin(t)+c and you should type (4cos(x),x,0.2 pi) into NINT.

Let's do another one!

Find the particles displacement for the given time interval.
Given v(t)=.25t^3-t, 0 is less than or equal to t which is less than or equal to 3.

For this problem, you would take the same steps as above.

Another week, another blog post.

The week started off with a quiz over 7.1-7.2. I felt ok about it. There was a few that I just didn't know how to do and totally guess because I was getting frustrated, Other than that, I felt pretty good about it.

We then moved onto section 7.3 and spent the rest of the week on it. Section 7.3 is all about volumes of solids of revolution and finding them using the disk or washer method. The disk method is used if the axis of revolution is the boundary of the plane region and the cross sections are taken perpendicular to the axis of revolution. The volume equation  is taking the integral from a to b of the function squared multiplied by pi.

The washer method is used if the axis of revolution is not a boundary of the plane region and the cross sections are taken perpendicular to the axis of revolution. To find the volume of this you take the integral from a to b of the top function squared minus the bottom equation squared multiplied by pi.

The concepts were easy to understand and I flew through the homework. I like this stuff. The only thing that really confused me was Austin rambling on about some way that sounds 10x more complicated than this. Overall, I feel really confident that I understand what is going on.

We ended this week on a quiz over the disk and washer methods and I felt like it went very well. Overall, all is good.

This week, we finished chapter 6 and moved on to our last chapter. It is kinda strange that we are almost done with learning new stuff but it will be very nice to have a long time to review.

We started off the week with chapter 6 review. We did something a little bit differently tho. Instead of having a homework assignment, we did stations around the room with different problems. I honestly didn't like this kind of review. I like taking my time with problems instead of having only so many minutes to figure it out and go onto the next one. I felt rushed and didn't really enjoy it. I also felt like I would have had a better review if I had done the review assignment instead. Just my opinion.

Then we took the chapter 6 test. Dun dun dun! The multiple choice went well I thought. I really understood this chapter so I felt really good going into this test. The one problem I had was with the related rates problem and looking back I just did a stupid. Overall, I think I aced the multiple choice test. The short answer on the other hand was a struggle. I don't know why it was. Maybe I was making it more complicated than it needed to be. Maybe my brain just went "Haha nope" and fried out on me. I think that I will end up doing my first mastery on that test. That is how not confident I feel about it.

After the chapter 6 test, we moved onto chapter 7. It's kinda stuff we have seen before, which is nice. It's all about position, velocity, and acceleration and that's not bad at all. You just have to really think about what the problem is asking and you are good to go. So far, I don't think this chapter will be hard or bad so yay.

Until next time!

It's always a good week when it starts out with a snow day :)

We started the week off with a 6.1-6.2 quiz. I felt pretty good about it. So far, I feel pretty confident with Chapter 6. It seems like stuff we have done before but taking it to the next level. The hardest part is remembering stuff from algebra. But once I remember all that jazz, I'm pretty good with the rest of the material.

We continued on with Chapter 6 by doing section 6.4. Again, it is stuff we have done before but with a twist. 6.4 is was separation of variables. It sounds exactly what it is. You find the original equation by putting the same variables on the same side and integrating. What I mean by that is that you are given an equation like dy/dx = x/y You want to put the dy and y on the same side and the dx and x on the other side. Once you do that, you integrate and find the y= equation. Waalaa. It's not hard or mind bending. We also looked at problems with exponential growth. The same ideas but with exponential growth. In this section we also looked at radioactive decay. This uses the same concept of exponential growth so nothing too different. After learning 6.4, I felt really good about Chapter 6. I really understood the material and only skipped a few on the homework assignments compared to skipping a lot like I usually do. I feel that I will do very well on the upcoming Chapter 6 test.

On Friday, we did and FRQ. This was kind of intimidating because this is stuff on the AP test and the AP test is super important. This FRQ wasn't bad at all. I just blanked out on everything we have ever learned so that was fun. But going over it together, I realized that this wasn't that hard. You just have to remember every little step because if you screw up even a little, it affects your answer a lot. 

One more week done. Yas! 

This week we started Chapter 6: Differential Equations and Mathematical Modeling. Sounds fascinating I know. But Section 6.2 was just review for us. We did U-substitution back in the fall. I had a rough start to this section; I forgot how to u-substitute. But when doing the homework, I got back into the grove and everything came back. We used this knowledge to u-substitute in the context of definite integrals. I was super confused while doing the exploration but when the idea was explained to me and I saw an example, it made sense

Sorry for how late this is. But yay snow day!

I honestly had a little trouble with section 5.1 but only because I didn't math at all during winter break. But after a day or two it all came back, so we are all good.

The next two sections were not bad at all. I understood them very well and I feel that I could do well on the next quiz. Also my notes look all smart and whatnot. 

The first part of section 5.3 was all about the rules of definite integrals and the mean value theorem for definite integrals. Sounds brutal, but most of it is just common sense. Like I said before, I feel very comfortable with these ideas and concepts so yay Chloe!

The second part of section 5.3 was all about the anti-derivatives of definite integrals. Again, really not that bad and common sense. It all makes sense to me. That is the important thing here right? 

I am kind of nervous for the AP style quiz because I really don't know what to expect. How much of it is it set up like Cresswell's quizzes? AGH. I glad we are starting to see more AP style things because the test will come here faster than we expect. We will see what happens.

Until next time!