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Teachers & SchoolsStudy Guide

In this section we left the integrating to the calculators. That's practical, because calculators can integrate much more quickly and accurately than we can. Instead, we've been spending our time practicing imagination.

The word "imagination" might seem out of place in the "Practical" section of this lesson. For some reason, imagination often gets cast as the villain alongside the hero of practicality. We're not sure why, because imagination is terribly practical.

Imagination means "forming a mental image of something not present to the senses." If you're listening to the sound of the car and trying to envision which part of the engine is broken, that's imagination. If you're preparing to perform surgery and thinking about where the organs lie under the skin, that's imagination. If you're preparing for a race by visualizing yourself running it perfectly, that's imagination.

Imagination means "the act of forming a mental image of something, never before wholly perceived in reality." If we want to design or make anything, anything at all, we need a working imagination. What are you interested in? Buildings? Cars? Shoes? Three-dimensional puzzles? In order to create something new, we need to be able to see something that doesn't yet exist.

When we practice something, we get better at it. Holding weird 3-D objects and their various slices in our head will make us that much better at everything else that requires imagination, which is just about everything. Now that's practical.

### I Like Abstract Stuff; Why Should I Care?

Most calculus courses are very big on the idea of limits. In particular, the books are adamant about defining an integral as the limit of a bunch of finite sums.

This isn't how the development of calculus actually went. Integrals came before limits. The history of calculus goes more like this:

*Newton and Leibniz:*Hey, let's chop up a region into infinitely many rectangles that are infinitesimally thin, then add up their areas to find the area of the original region! Infinitesimal quantities are neat!*Other Mathematicians:*Um... what's an infinitesimal? Any real number that small must be 0. Great ideas, guys, but we need a way to make this more rigorous. How can we do math with things when we don't know what they are?*Later Mathematicians:*We can use limits to make that calculus stuff more rigorous. This is how everyone should do it from now on.*Abraham Robinson*: Actually, Newton and Leibniz had the right idea. We can make infinitesimals rigorous after all.*Ghosts of Newton and Leibniz*: Hah. Told you so.The point is that it's perfectly fine to think of an integral as an infinite sum of infinitesimally skinny things. That's how integrals were developed in the first place, and for some people this idea makes a lot more intuitive sense than the "limit" definition. It's possible to do calculus without having limits at all, but this idea hasn't caught on in most schools.

You can also think of integrals as a way to multiply two numbers where one of the numbers is changing. When the numbers are constant we have the formula

distance = speed × time.

When speed is changing, we have

### How to Solve a Math Problem

There are three steps to solving a math problem.

1) Figure out what the problem is asking.

2) Solve the problem.

3) Check the answer.

### Sample Problem

A conical water tower is 20 feet tall and 14 feet in diameter at the top. The tower starts completely full and then half the water drains out. To the nearest foot, what is the depth of the remaining water?

Answer.

1) Figure out what the problem is asking.

We're told there's a conical water tower and given its dimensions. We really need the radius, not the diameter. The tower starts out full of water, so the volume of water is equal to the volume of the tower. After half the water drains out, the volume of water remaining will equal half the volume of the tower.

The question is at what depth,

*H,*the volume of water equals half the volume of the tower.

Take horizontal slices of the tower, using

*h*for the distance from the bottom of the tower to the slice. Then the volume of water in the tower is

We want to find the value of

*H*for which2) Solve the problem.

We need to get

*r*in terms of*h*so we can do the integrals. Slice the cone down the middle to see the similar triangles.We have

so

The volume of the tower is

So half the volume of the tower is

We need to find

*H*such thatWhen we evaluate the integral on the left-hand side of this equation, we get

We set this equal to the volume of the tower and solve for

*H*.Rounding to the nearest foot,

*H*≈ 16.The water left in the tower will be approximately 16 feet deep.

3) Check the answer.

To check the answer, use a calculator to evaluate the volume of the tower:

Then use the calculator to evaluate the volume of water when it's 16 feet deep:

Since 525 is approximately half the volume of the full tower (), our answer is reasonable.