Studying for a Test with Variable Answers

[stripedcat]

This technically might not belong here, mods, put it in it's proper place if not.

So I'm studying for a test on Monday. Not a big deal, we use webassign, you may commence with the horrified gasps and groans, I know.

So here's my odd little situation.

I get a problem like this one

Integral 0 to pi/2 3sin^2x cos^2x dx

The 3 there might be a 4, or an 8, or a 9, or whatever, but the rest remains the same.

Once you realize that so long as that's the only part that changes, you know the answer immediately. It's always whatever the number is, times pi, over 16.

3pi/16 in this case.

That's great and all, but in order to learn how to do the problem that's not terribly helpful. I can force my way through it but that's hard to do when you instantly know what the answer is going to be the second you look at it.

How do you study in a situation like this?

 

[Prove It]

This technically might not belong here, mods, put it in it's proper place if not.

So I'm studying for a test on Monday. Not a big deal, we use webassign, you may commence with the horrified gasps and groans, I know.

So here's my odd little situation.

I get a problem like this one

Integral 0 to pi/2 3sin^2x cos^2x dx

The 3 there might be a 4, or an 8, or a 9, or whatever, but the rest remains the same.

Once you realize that so long as that's the only part that changes, you know the answer immediately. It's always whatever the number is, times pi, over 16.

3pi/16 in this case.

That's great and all, but in order to learn how to do the problem that's not terribly helpful. I can force my way through it but that's hard to do when you instantly know what the answer is going to be the second you look at it.

How do you study in a situation like this?

How do you know just by looking at it that $\displaystyle \begin{align*} \int_0^{\frac{\pi}{2}}{\sin^2{(x)}\cos^2{(x)}\,\mathrm{d}x} = \frac{\pi}{16} \end{align*}$?

 

[stripedcat]

Funny you should ask because I just got that one. Literally.

I know because after doing about 10 practice problems, you see the trend.

It's always the number in front of the sin^2x, times pi, over 16.

In that instance its just 1pi/16 which naturally becomes pi/16. Even if you had no idea how to do an integral, if you saw five of these in a row you'd pick up on the fact that it's whatever that number is*pi/16.

My problem is learning it when it's that blindingly obvious. The first time I did it, obviously I had to go the long route, but after 2-3 times you kind of go 'wait a minute here...'

 

[Prove It]

Funny you should ask because I just got that one. Literally.

I know because after doing about 10 practice problems, you see the trend.

It's always the number in front of the sin^2x, times pi, over 16.

In that instance its just 1pi/16 which naturally becomes pi/16. Even if you had no idea how to do an integral, if you saw five of these in a row you'd pick up on the fact that it's whatever that number is*pi/16.

My problem is learning it when it's that blindingly obvious. The first time I did it, obviously I had to go the long route, but after 2-3 times you kind of go 'wait a minute here...'

Which means that you are learning. Seems like a pretty effective study technique to me...

 

[stripedcat]

Yeah but if anything ELSE changes... say, the terms of integration, or exponents are 4s instead of 2s...

I could change them and try to solve myself, but I'd have no way to know if I was right unless wolfram or something confirms it.

The advantage here is the ability to do slightly different problems over and over and over again... The disadvantage, well, you see what my problem is...

 

[Prove It]

Yeah but if anything ELSE changes... say, the terms of integration, or exponents are 4s instead of 2s...

I could change them and try to solve myself, but I'd have no way to know if I was right unless wolfram or something confirms it.

The advantage here is the ability to do slightly different problems over and over and over again... The disadvantage, well, you see what my problem is...

Then you think about what is similar about the problems, and what is new, and how you could modify what you have done to get over the new part...

 

[Deveno]

There is a theorem of calculus which goes:

If:

$\displaystyle \int_a^b f(x)\ dx = M$

then:

$\displaystyle \int_a^b c\cdot f(x)\ dx = c\cdot M$

Which is your "number in front" observation.

At some point, you learned (the hard way) that:

$\displaystyle \int_0^{\pi/2} \sin^2x\cos^2x\ dx = \dfrac{\pi}{16}$, and you are now applying the theorem above.

Now, you can just continue to "use this because it works", or you can investigate WHY it works. Your call.