And we want to add this squared.

Now we know what ds is. ds is equal to the square root of dx dt squared plus dy dt squared times dt.

I really encourage you to take out some nice clean graph paper, and plot the points exactly and then you can even try points like negative 1 2 and positive 1 2 and you'll see that the curve shapes out like this.

So 4x squared minus x squared is 3x squared.

And in the denominator what happens? This 3 and minus 3 cancel out.

Let's start to answer this question with a math problem. I know how to calculate the area of a circle.

So let's evaluate it at r. So this is pi r squared, and then for x, we'll substitute the positive r times r minus pi x cubed, but now we have this r here, so r cubed over 3 minus pi r squared, and then we have a minus r here, because we're evaluating the antiderivative at minus r, times minus r, minus pi minus r cubed. So what's minus r cubed?

2 squared, x squared, so 4x squared times 2 is going to be 8x squared.

And then, we want to add this squared.

So 15 squared plus 20 squared is going to be equal to r squared.

So the area is going to equal pi times this squared. So if you square this quantity, you just get rid of the square root sign, right? So pi r squared minus x squared, and that's the area, and so what's the volume of that disk?

It tells us that this side squared, so let's say x squared, plus this side squared, plus x squared is equal to the hypotenuse squared.

This distance squared plus this distance squared is going to equal that distance squared.

So we know that a squared plus b squared is equal to c squared.

Minus sine of t squared plus the cosine of t squared, this is equivalent to sine of t squared plus cosine of t squared.

This is a squared minus b squared.

4x squared is 2x squared.

The distance squared is going to be equal to this delta x squared, the change in x squared plus the change in y squared.

So we could say (a) plus (b), the whole square's area, is (a) plus (b) squared. ..(a) plus (b)... ...squared, and that's equal to (a) squared plus 2ab plus (b) squared.

This x squared cancels out with this minus x squared.

So this will simplify based on this pattern right here to a squared, which is 2x squared, minus b squared, minus 8 squared.

It's x squared plus y squared is equal to r squared.

49x squared is a perfect square.

49y squared is a perfect square.

Plus 3 squared 3 squared is 9 sine of 9.

let me write h squared that's different shade of yellow so we have h squared is equal to sixty five.

That equals a squared minus b squared.

We have 70a 2, 15 or negative 15a 2.

So plus 11 squared is equal to d squared.

If you really wanna to verify it for yourself,

So what's threei squared? threei squared is equal to three squared times i squared.

The limit as x approaches infinity of 3x squared plus x over 4x squared minus 5.

Now 1 is just cosine squared over cosine squared.

So we can see that if we take 15 squared, that's one of the short sides, I'm squaring it. And then add that to the square of the other shorter side, to 20 feet squared.

This side squared plus that side squared is equal to hypotenuse squared, because this is a right triangle.

This gets us 2 squared is 4 times x squared minus 8 squared, minus 64. And we're done!

Well the 4t squared is a squared.

Because if you have minus 2y squared is 4y squared.

Was y squared over 2, minus y squared over 2.

4a squared times 3a is 12a cubed.

We need to factor 49x squared minus 49y squared.

And you're just going to be left with an a squared minus a b squared.

4 pi minus x squared minus 4 pi x squared.

It's going to be the 6.37 squared times 10 to the sixth squared.

So you could say it's equivalent to 49 times x squared minus y squared. And you say, oh, this fits the pattern of this is a squared minus b squared.