Finding volume of enclosed regionThe base of S is the region enclosed by the parabola y = 9 − 9 x 2 and the X - axis. Cross-sections perpendicular to the X - axis are isosceles triangles with height equal to the base.

Question

Finding volume of enclosed regionThe base of S is the region enclosed by the parabola   y  =  9  −  9      x    2   and the X - axis. Cross-sections perpendicular to the X - axis are isosceles triangles with height equal to the base.

Julianna Bell · Accepted Answer

Step 1You want to take the volume as   V  =      ∫    a    b    A  (  x  )    d  x  ,, where A(x) is a typical area slice. I think from comments you can see that   A  (  x  )  =      1    2    B  H  ,, where B is the base of the triangle and H is the height.Step 2Thus   A  (  x  )  =      1    2        (    9    −    9          x      2        )        (    9    −    9          x      2        )    ..It helps to sketch these things to see what is going on. Learn how to sketch 3 dimensional solids and your life will be much more enjoyable. Next you want the limits of integration. They are the the zeroes of the function y, this being where the function meets the x axis, and trivially found. See if   V  =      ∫          −      1              1            1    2              (      9      −      9              x        2            )        2      d  x meets with your mathematical sensibilities.

The base of S is the region enclosed by the parabola y=9-9x^{2} and the X-axis.

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