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# Volume with cross sections: intro

AP.CALC:
CHA‑5 (EU)
,
CHA‑5.B (LO)
,
CHA‑5.B.1 (EK)

## Video transcript

you are likely already familiar with finding the area between cars and in fact if you're not I encourage you to review that on Khan Academy for example we could find this yellow area using a definite integral but what we're going to do in this video is do something even more interesting we're gonna find the volume of shapes where the base is defined in some way by the area between two curves and in this video we're gonna think about a shape and I'm Li draw it in three dimensions so let me draw this over again but with a little bit of perspective so let's make this the y-axis so that's the y-axis this is my x-axis that is my x-axis this is the line y is equal to six right over there y is equal to six and this dotted line we could just draw it like this and so this would be the point x equals two and then the graph of y is equal to four times the natural log of 3 minus x would look something like this looks something like this and so this region is this region but it's going to be the base of a three dimensional shape where any cross section if I were to take a cross section right over here is going to be a square so whatever this length is we also go that much high and so the cross section is a square right over there the cross section right over here is going to be a square whatever the difference between these two functions is that's also how high we are going to go this length which is six at this point this is also going to be the height it is going to be a square it's gonna be quite big we have to scroll down so we can draw the whole thing roughly at the right proportion so it looks something like this this should be a square that's gonna look something like this and so the whole shape would look would look something like this would look something like that and try to shade that in a little bit so that you can appreciate it a little bit more but hopefully you get the idea and some of you might be excited and some of you might be a little intimidated well hey I've been dealing with the two dimensions for so long what's going on with these three dimensions but you'll quickly appreciate that you already have the powers of integration to solve this and to do that we just have to break up the shape into a bunch of these you could view them as these little square tiles that have some depth to them so let's make that into a little tile this one into it that also has some depth to it you could even I could draw multiple places you could view it as a break it up into these things that have a very small depth that we could call DX and we know how to figure out what their volume is what would be the volume of one of these things well it would be the depth times the area times the surface area of this cross-section right over here moved in a different color so what would be the area that I am shading in in pink right over here well that area is going to be the base length squared what's the base length well it's the difference between these two functions it is going to be 6 minus our bottom function is 4 times the natural log of 3 minus X and so that would just give us that length but if we square it we get this entire area we get that entire area you square it and then you multiply it times the depth you multiply it times the depth now you have the volume of just this little section right over here and I think you might see where this is going now what if you were to add up all of these from x equals 0 to x equals 2 well then you would have the volume of the entire thing this is the power of the definite integral so we could just integrate from x equals 0 to x equals 2 from x equals 0 to x equals 2 if you drew where these intersect our base you would say all right this thing right over here would be this thing right over here where it's DX instead of just multiplying DX times the difference between these functions we're going to square the difference of these functions because we're visualizing this three-dimensional shape the surface area of this 3 dimensional shape as opposed to just the height of this little rectangle and if you were to evaluate this integral you would indeed get the volume of this this kind of pedestal horn look thing this is not an easy definite integral to evaluate by hand but we can actually use a calculator for that and so we can hit math and then hit choice number nine for a definite integral and then we just have to input everything we're going from zero till two of and then we have let me open parenthesis cuz I'm gonna square everything six minus four times the natural log of X or actually the natural log of 3 minus X and so let me close the parentheses on the natural log part and then if I close the parentheses on this whole thing I want to then square it and then I'm integrating with respect to X enter I got approximately twenty six point two seven so approximately 26 point two seven and this is a volume here so if we thought about units it would be in our units cubed or cubic units
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