05Volumes of Revolution2 - write: 3 3 1 3 1 3 1 2 = -= = =...

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V OLUMES OF R EVOLUTION If a curve with fixed boundaries is rotated around the y-axis, a 3-dimensional solid is formed. For example: Take the function x y = , 1 0 x . The volume is the product of the area times the height. = × = h A V (area of a disk)x(height) Each circle has radius y as x goes from zero to 1. The area of one circle (disk) is 2 y π . The width of each circle is the change in x from one circle to another. Therefore, we can sum the areas of the circles using the formula: = × = n k x x A V 1 ) ( , where n is the number of circles and x is the width of each circle. If we allow the number of intervals to increase, the width of the intervals to approach zero, then the limit of this sum as 0 x is: If the curve is rotated around the x- axis, a cone is formed. We can find the volume of this shape by slicing it into cylindrical sections (circles) and summing their areas. x y = [ ] = = b a b a dx x f dx x A V 2 ) ( ) (
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For each disk, the area is 2 y π , but using the function, we know that x y = , therefore we can
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Unformatted text preview: write: 3 3 1 3 1 3 1 2 = -= = = x dx x V . Notice that we got the volume of the shape to be h r 2 3 1 , which happens to be the equation for volume of a cone. A solid of revolution is one generated by rotating a plane region a out a line that lies in the same plane as the region. For example, if the line 3 = y , 4 2 x , is rotated about the x-axis, what solid is formed? 3 = y 2 4 ( 29 18 2 9 ) 2 4 ( 9 9 9 3 ) ( 4 2 4 2 4 2 4 2 4 2 2 2 = =-= = = = = = x dx dx dx y dx x A V Ex . Write the integral for the volume of the plane region defined by the function r y = from x=a to x=b Ex . Find the volume of the solid obtained by rotating about the x-axis the region under the curve x y = from 0 to 1. Assign: p.456 #1, 7, 9, 13, 27, 29...
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05Volumes of Revolution2 - write: 3 3 1 3 1 3 1 2 = -= = =...

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