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AP Calculus AB

An interactive log for students and parents in my AP Calculus class. This ongoing dialogue is as rich as YOU make it. Visit often and post your comments freely.

Wednesday, March 08, 2006

The amount of WORK

Yes calculus takes a lot amount of work. Imagine our brain cells, constantly moving the data through the brain. Our brain only has a particular volume to it, but it takes a lot of force to move all the data swimming inside it to a particular destination.

*hehe* Just kidding guys.

But really the topic we covered on Tuesday was on WORK. To those who took Physics, you may already have conjured up the formula for work:


Applying Calculus…

Overlooking the Problem from Tuesday…

We are given a rectangular prism tank 6 ft long, width of 5 ft, and height of 4 ft. On the top of it is a spout 2 ft. tall. The prism is filled with a liquid whose density is 40 lbs/ft³. Write the integral on amount of force it takes to move the water out of the tank.

When a picture is not provided, it is always good to draw one to understand the problem more.

To start off the problem:
1) Draw a picture, unless it is provided.
2) Write down what you are given.
3) Recall a formula.
For this problem, it’s the WORK formula. Work = F x D

In the problem we are not given the force straight off the bat. We are looking for the force of gravity, which is measured in lbs. We are given that the density in the units lbs/ft³. What do we multiply the density by to leave us off with just lbs.?

Lbs/ft³ x _____ = lbs.

The answer is: multiply it by ft³. In other words we multiply by the volume.

To find the volume of a rectangular prism: (length) (width) (height)

We are to move all of the water out of the tank. Taking an arbitrary slice, the height of the slice will be a very small itty bitty change in y.

We all know that it’s dy.

The length and width is given in the problem 6 ft. and 5 ft.

Good Job. You have found the weight of the substance. Now for distance:

Looking at the prism, imagine an arbitrary slice on the surface. That slice has 2 ft of distance to move out of the tank. Now imagine an arbitrary slice on the very bottom. The height of the prism add the height of the spout gives you 6 ft of distance to cover to get out of the tank. What equation would give you the possible heights?

Answer: 6-y

One thing left to figure out, is the limits of integration. The problem had said the prism is filled with a liquid substance. The height of the prism is 4 ft. So therefore, the limit of integration is 0 to 4.

Now to write the integral it is:

On Tuesday we were given another similar problem but this time it was a cylinder. The only thing that changed was how to find a volume of a cylinder. I guess this is where our bank of volume formulas comes to play once again. It is always good to have any of our formula sheets around.

So it was another day in calculus. I believe we have only 18 classes to go? Or maybe even less before the AP exam. Not to scare anyone away, its one tough exam. Everything Mr. K says is true. It’s nothing like any of the exams we had ever written. Good practice, a good understanding of the concepts is needed, and a great memory too. Take advantage of all resources available to us. We’ll need it.

Well see you all tomorrow =)

Reply to COMMENT: I find the quizzes a good review of the topics we've already covered. It also sets us ready for the time given in the exam to answer multiple choice questions. It also helps us not to be so dependant on the calculator. So overall, its helps us get ready for that exam on the beginning of May =)

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