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Find the volume of the solid whose base is the region bounded by the lines x 4 y 4 x 0 and y 0 if the cross sections taken perpendicular to the x axis are semicircles.
Volume by cross section. V int limits a b a x dx for example suppose we want to find the volume of the solid with each cross section being a. The volume of a solid with known cross sections can be calculated by taking the definite integral of all the cross sections with a x being equal to a single section. Volume left limit right limit area of cross section at x d x where in whatever manner we describe the solid it extends from x left limit to x right limit.
The volume v of the solid is. We must suppose that we have some reasonable formula for the area of the cross section. For example let s find the volume of a solid ball of radius 1.
The area a of an arbitrary square cross section is a s 2 where. This cross section if we re looking at it at an angle and if the figure were transparent it would be this cross section right over here. It would be that cross section right over here which is a semi circle.
Finding the volume of a solid find the volume of a pyramid with a square base of side length 10 in and a height of 5 in. The volume by cross section method takes the area of all of the slices of the shape and adds them together to find the total volume. If we were to take this cross section right over here along the y axis that would be this cross section.
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.
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Using The Cross Section Method Find The Volume Of The Solid Obtained By Method Cross Section Volume
