Volumes of Regions with Known Cross Sections

Find the volume of the solid bounded by the curves y = x^2 and y = 8 - x^2 who cross sections are perpindicular to the x - axis with one side on the xy plane using semi circle cross sections

Ok so i have to find total volume which is equal to the Rheiman sum of the the volume of the semi circle which is



and where delta thickness is the change in x?
and the radius of the semi circle is 1/2 the height between y = 8 - x^2 and y = x^2
which gives





then you will have the intergral from x = -2 to x = 2 because this is where they intersect



For some reason I dont think im accurately doing this

Quote:

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Originally Posted by RockHard

Find the volume of the solid bounded by the curves y = x^2 and y = 8 - x^2 who cross sections are perpindicular to the x - axis with one side on the xy plane using semi circle cross sections

Ok so i have to find total volume which is equal to the Rheiman sum of the the volume of the semi circle which is



and where delta thickness is the change in x?

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Yes, because your cross sections are "perpendicular to the x-axis", the thickness, perpendicular to the area, is measured along the x-axis.

Quote:

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and the radius of the semi circle is 1/2 the height between y = 8 - x^2 and y = x^2
which gives





then you will have the intergral from x = -2 to x = 2 because this is where they intersect



For some reason I dont think im accurately doing this

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Two errors:
You left out the "thickness" which becomes "dx" in the limit. NEVER write an integral without the "dx"!
Also, it is the entire polynomial that is squared. .

Now, what are the limits of integration?

The limits of integration are lower limit = -2 and upper limit = 2 because thats where the two graphs intersect?

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