WEBVTT
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we're finding the volume of the solid obtained by rotating
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the region. Bounded by Why score equals X X
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equals two. Why rotating about the y axis?
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So let's first start off by station in the region
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. So this is the X and Y axes.
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We can first start off by doing that. Why
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squared equals X. So the wind was X squared
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Line typically looks like this. Yeah, but when
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we take the inverse of that, we basically you
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can't look like this because no matter what the value
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of why is X is always positive. We're basically
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rotating by ninety degrees clockwise, and then we can
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do the what X equals to wildlife. So normally
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X equals Y looks like this, but we're stretching
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it out a little bit more, you know that
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. So now when you find the two points of
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intersection So the very clear one is your zero and
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the other one we can solve by recording these two
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equations together. So we get why squared is equal
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to two. Why? And the very obvious answer
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here is that y equals two. Because if why
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times wise equal to two times alive. And why
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has to be two. And if we plugged that
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in and we get X equals four. So this
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point right here where the two lines intersect is for
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two. We're voting this all about the y axis
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for here. Oh, this is the region that
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we care about. So now when you draw out
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, the typical Washington is created, and so it
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looks like a washer. A Samantha circles attracted out
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from the marker circle. So what do you find
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? The cross sectional area that we find in terms
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of why this is Why changes the area changes.
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The bigger circle is caused by this line. Right
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here is that school is always the exam is always
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greater, creating the larger circle. So that's X
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equals to why we plug in, um, the
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axe value because the X value creates the is the
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radius of the circle. So the big R is
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just too wise, and so therefore the smaller has
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to be just y squared. And if we finally
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simplify all of this, you get that's equal to
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pi comes for squared. Mine's wants before and so
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that is our cross sectional area function. We can
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now go ahead and proceed with our interval. So
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as I mentioned before, for two is the point
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of intersection along with origin. So we're going from
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Wankel zero to y equals two. When we put
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in the cross sectional area function that we found before
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running merely plowed the part and then was left is
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for my square minus y to the fourth d Y
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, and we can find this too anti differentiation.
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We said that this is pies the anti derivative of
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four. Why square is just four times why three
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to over three minus y five over five. That's
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the anti driven apply to the forth that is a
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valid way to add zero and two. We can
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plug this end. We get it. Pi times
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for two to the three is eight three minus two
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to the fifth is thirty two or sixteen? No
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, actually, it's Terry to five. We don't
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care about the case where X equals zero or Y
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equals zero, because it just gives us hero.
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So this is our final answer. We multiply all
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this out when you finally get the answer. That's
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this Entire Integral is equal to sixty four pi over
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fifteen