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Calculate the volume of the ramp in Figure 17 in three ways by integrating the area of the cross sections:

(a) Perpendicular to the x-axis (rectangles)
(b) Perpendicular to the y-axis (triangles)
(c) Perpendicular to the z-axis (rectangles)

  • Calculus - ,

    We do not see the figure, but could probably help anyway.

    Along whichever axis you do the integration, the method is to cut up the volume into slices perpendicular to the x-axis, say. Each slice is of thickness dx and the width and height will be a function of y and z, which should in turn be transformed to a function of x. Do the integration along the x-axis and find the volume.

    You can proceed in a similar way along the two other axes. The resulting volumes should, of course, be identical.

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