How do I find the volume?

I don't follow where you got 1/2 2y or any of that at all really.

@justanguy The area of a triangle is half the product of its base and height. I substituted the base of $2y$ and the height of $2y$. I then used the fact that the equation of the circle is $x^2+y^2=1$ to replace $y^2$ with $1-x^2$.

How did you get the base and height?

@justanguy The base in your example is the same as the base in the example from my screenshot. Do you understand how they got their base? Once we know the base, we automatically get the height as well, since we are given that they are equal.

How do we know they are equal? I dont understand what a cross section is/

@Adriana How do you get the base and height?

I really need help I can't do any of these, I injured my hand and can't type well and I have 8 hours of homewokr still.

@Adriana How do you get the base and height?

@justanguy I knew that the height equalled the base because (quoted directly from your question): "the cross sections perpendicular to the $x$ axis are triangles whose height and base are equal". Note that the pink triangle in Figure $13(c)$ is a cross-section. The line segment $AB$ is the base of the triangle and has a length of $y+y=2y$, so the height is also $2y$.

I don't know what that means for there to be invisible triangles in this problem, I can't conceptualize that into a formula/.

Why isnt this a cone?

Take a look at Figure 13(a). Do you see the pink triangle $ABC$? That's the triangular cross section we're looking for.

why a triangle?

From your question, it explicitly states that: "the cross sections perpendicular to the x axis are triangles". I didn't make it up; the question told us what shape to use.

But it has a base of a circle

so a cylinder with sloped sides

What do you mean by "sloped sides"?

the triangles

cut into the cylindar

to make it a cone with triangle sides

Ah, I see what you're thinking.

It turns out that EVERY cross section perpendicular to the x-axis will be a triangle whose height and base are the same.

Now let's think about a cone for a second.

i dont understand a cross section

Imagine a soft cake that is shaped like a cone.

Now imagine slicing it in half.

If you slice it right in the middle, then you'll get a triangular section. But what if you don't slice it in the middle?

I dont follow

in half vertically?

Yes.

See the orange hyperbola?

Slicing the cone vertically won't give you a triangular cross section. The sides will be curvy (we call it a hyperbola).

yes

So it won't really look like a cone. The sides should be more straight.

It will look more like this:

I am so fucked I have class in eight hours and I havent started the homework yet

I get it now

I thought crosssection was a shape of the object

Cool. Anymore questions?

well i dont know what to do still

Then tell me what you know so far.

the original problem

I get the circle for the base

but where do i go from there

The key to finding the volume of a 3D solid is to figure out its cross sectional area.

If we can come up with some function A(x) that represents cross sectional area, then that's half the battle.

but i dont get the problem

I have a base of a circle

what makes the shape what it is

To understand the problem, you only need to look at a cross section.

oh

so I will forget the rest for now

the cross section is a triangle

so I know to work with 1/2bh

Good. So we need to figure out its base and height.

but how does that give me the area of this cylindar thing

Let's start with just the base.

Here's something to memorize.

the base is unknown for now right?

Volume = integral of cross sectional area

That's why we care so much about the cross sectional area.

I get that but it is the volume of a triangle not the cyclindar thing

It is a weird 3D shape. Not quite a cylinder, not quite a cone.

but it isn t important toknow what it is for sure is it?

just the crosssection?

Figuring out the cross section stuff completely defines how the 3D shape looks.

It's all you need to know.

That's what's cool about calculus.

but the object gets smalled with different depths

how do those get figurd into the area

i know how they work

It's true that the cross sectional area isn't constant.

It will depend on the "depth", or in this case, x.

That's why we want to come up with a function, A(x), that depends on x.

yeah but how does this depend on it

the triagle

Alright, so let's slice the 3D shape at some distance x along the x-axis.

Let's think about the base length of this slice.

I get it

What will it be?

I just need to work with a triangle

2*radius

2?

The base length is not constant.

It depends on where we slice it.

Think about the endpoints of the slice.

the largest base is 2?

Yes, the largest base is 2.

so similar triangles

Now let's think about the endpoints of the slice.

Suppose the point where it touches the upper semicircle is (x, y).

Then the point where it touches the lower semicircle is (x, -y).

With me so far?

no

the triangle/

?

See the pink triangle ABC?

yes

I'm focusing on the points A(x,-y) and B(x,y).

x and y are hard to see here

if I have a 2d triangle where is it?

We're only looking at the base of the 2D triangle.

It's a straight pink line, AB.

well I would use a similar triangle

whihc would give 2/1 = (2-y)/x

y being the height and x being the width of the base at that height

Can you explain what the (2-y) represents?

2 is the height of the triangle

and y is where you are on it

Hmm that's not quite right.

Here's how I would do it.

my answer is wrong

We want to find the length of the base (the length of AB).

Can you see that the length must be 2y?

no

Can you see that the points are A(x,-y) and B(x,y)?

no

I dont nkow what A and B are

Can you see points A and B on the diagram?

yes

but adding x and y dont make sense to me

Can you see that the points are A(x,-y) and B(x,y)?

oh

I guess yeah

Can you see that the length must be 2y?

no

where is 2 from

the max length is 2 I though

Can you see that the distance from A to the x-axis is y?

Or can you see that the distance from B to the x-axis is y?

yes

So we just add them together. y + y = 2y.

Does it make sense that the length of the base is 2y?

yes

but we know that the base is 2

right?

so it is 1

No.

It's 2y.

If y = 1, then the base is 2(1) = 2.

I see that

I am not sure I undersatnd the idea of 2y

y is the y axis?

y is the vertical distance from the x-axis to the point B.

B or C?

B is on the x axis isnt it?

like in my drawing

the triangle has the x axis in thecneter?

I can't imagine what that graph means

it doesnt make sense to me

See the picture on the right? A and B aren't on the y-axis.

why is the triangle going past the y axis like that?

is the x axis cutting through it kind of in the corner>

I dontunderstand what it is supposeto represent

I don't understand your question. Can you elaborate?

that pciture doesnt make sense

I am too tired to doth si

my hand is injured

Why doesn't it make sense?

and i cant type much anymore anywyas

the x and y lines are at a weird angle

and the triangle is going through them oddly

It's a 3D shape. It's drawn from a perspective.

so what is 2y

Remind me what y represents.

the length?

or the hieght

I have no idea

I am defining the point B(x,y) to be the point where the triangle intersects the upper semicircle.

So y represents the vertical distance from point B to the x-axis.

so its max hiehgt is b xy

and then so y is height

We are only looking at the base of the 3D solid. No heights yet.

oh

well they are all the same right?

equal triangles

The triangles have different sizes.

So does 2y make sense yet?

no

I dont undersatnd why we are doing this instead of simialr triangels

Similar triangles is more complicated. If we use similar triangles, what are the dimensions of your two triangles?

I am not sure how either method works

I dont understand the idea of interescting the upper semicircle

Which part?

I cant visualize the shape

so looking at it from the front b hits the semicircle on the right

A semicircle is a curve. The base of the triangle is a line. When a line intersects a curve, it makes a point.

ok

I am too tired for this Ith nik

thanks for the help though

I need sleep so I can go to class tomorrow and fail the quiz and turn in no homework