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## AP®︎ Calculus AB (2017 edition)

### Course: AP®︎ Calculus AB (2017 edition)>Unit 11

Lesson 5: Volume: known cross sections

# Volume with cross sections: semicircle

The volume of a solid with semi-circular cross sections and a triangular base.

## Want to join the conversation?

• I still don't get how 1-x is diameter. Isn't it suppose to be radius?
• I was confused for a while too because I thought this triangle was rotated around the x-axis to create a half cone. Don't understand why he used this shape as an example
• In the beginning of the video, Khan explained that the cross section that is perpendicular to the x axis will form a semi circle, but I am still having trouble seeing that a semi circle will be the 3D cross section for this figure. Why can't it be a triangle or a square?
• I think the semi circle is just an arbitrary shape that he has chosen for this example. You could make the cross sections triangular or square if you wanted to and still use the same base shape. That would just be a different example of the same general principle.
• Yeah I still don't get this😤
• Think about it in terms of Riemann sums, not cones.
• is the volume of a cone not 1/3 pirh? When you solve shouldn't you get 3 in the denom not 4?
• I thought the same thing as you at first. But after reading some comments I realized this is NOT a cone shape. In a perfect cone the cross sections are circular only when the cross section is taken PERPENDICULAR to the center line of the cone. In this video the semi circular cross sections are not perpendicular to the center line but perpendicular to the lower edge of the shape (represented by the x axis). This is only clear once you pay attention to the yellow lines in the left and right drawings. Then it becomes clear that 1) the y value is the diameter, 2) the x axis is the lower edge and 3) the yellow line is the upper edge of the base that lies flat/horizontal. The center line of the shape on the right would run from the points (0,.5) to (1,0).
• You say that y = 1 - x = Dia. or 2r ? If you view three dimensional view I see that 1 - x = y = radius and not dia. or 2 r. Everything else is well explained and understood. Please help me to understand that y = Dia. and not radius. If I would be able to see that y = dia. then radius is (1 - x)/2.
• 1-x is the base the semicircle (the diameter), not it's height (the radius).follow the yellow line it will be clear to you then.
• I've seen a couple of people explain that "1-x" is the "base (diameter)" of the semicircle, and therefore NOT the radius. I apologize for a redundant question, but I don't understand why. Visually, it seems very clear to me that "1-x" is half the diameter of a full circle "D/2", which would make "1-x" the radius. Doing "(1-x)/2" makes it seem like you're actually taking half of the radius. Could you help clear this up?
• I looked at the video and see why you were confused. Initially I did visualize the "slice" from the x axis to the line `y = 1 - x` as being pivoted from the x axis out into the pos and neg z direction to fill out 3-space. But I looked at the tent-looking diagram of rotation effect again and realized this critter lives inside of one octant - positive x's positive y's and positive z's. Look again at the tent diagram and notice where the x and y axes are and it might pop at you rightly too. we were both thinking of this as a rotational solid. In a sense it is. The rotational axis would be a median of the right triangle - the one that goes from the vertex on the x axis to the midpoint of the side on the y axis.
• This rotation thing always rotate my head
• Right at the beginning at , when Sal encouraged us to figure out the volume of the shape, I used a different method that still gave me the correct answer. Can anyone please tell me if the method I used (shown below) is wrong?

I first recognized the shape to be half a cone. The volume to calculate the volume of a cone is (pi*r^2*h)/3. To get half a cone, I divided that by 2, giving me the formula ((pi*r^2*h)/3)/2. From here on, I only need to substitute in the values for r and h to get the correct answer.

To get the height, I saw that in the diagram of the cone, you can see that the height of the cone is the same as the one of the bases of the triangle formed by the function f(x) in the first quadrant. Since the formula for f(x) is x+y=1, I figured the base had to be 1.

To calculate the radius, I noticed that another base of the triangle formed by the function f(x) in the first quadrant was the diameter of the cone. Again, since the formula for f(x) is x+y=1, I figured the base diameter had to be 1. Radius = diameter by 2. Radius = 1/2.

Substituting numbers back into original equation:
=((pi*(0.5)^2*1)/3)/2
=((1/4)pi /3) /2
=(1/12 pi) /2
= 1/24 pi
= pi/24.

Can someone please tell me if this method is wrong and if so, what is my error?
• It is wrong because you made assumption for the shape, and the height. Just because a diagram APPEARS to be a certain way, does not mean it is, you need to follow the information given (formula, labelled heights, lengths, etc).

So yes, it was wrong, because it will not always work unless the given information fits your assumptions, which may very well happen, but why chance it?