lecture04 - 1 Volumes Finding volumes is very similar to...

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Unformatted text preview: 1 Volumes Finding volumes is very similar to finding areas. With areas, we approximated the region by shapes we did know and then made the approximation better. With volumes for regions we don’t know, we do the same thing. To start, the area of general cylinder is Ah where A is the area of the base. What we do with volumes is take cross-sections of the region and look at really small cylinders that have the cross-section as a base called slabs. We add up all these slabs and this gives an approximation to the volume. Just like with area, we let the number of slabs go to infinity and we get the volume. *Draw a picture But what is the volume of a slab? Well, it’s a cylinder, so the volume is Ah . What we do is first break up the interval into subintervals [ x i ,x i +1 ] and take the base of the slab to be at the left end point. Let A ( x ) be the area of a cross-section at x . Then the volume of the slab is A ( x i )( x i +1- x i ) = A ( x i )Δ x . So the approximation we get is V ≈...
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