Sections of this figure, that our vertical, I should say our perpendicular to the X axis, those cross sections are going to be isosceles right triangles. Sections of the figure, that's what this yellow line is. What I've drawn here in blue, you could view this kind of the top ridge of the figure. Lets see if we can imagine a three-dimensional shape whose base could be viewed as this shaded in region between the graphs of Y is equal to F of X and Y is equal G of X. Your bounds should obviously be the least and greatest x-values that lie on the circle. You should have the base length from the previous step, which is all you need to find the cross-sectional area.Ĥ. The cross-section is an equilateral triangle, and you probably learned how to calculate the area for one of those long ago. Remember that to express a circle in terms of a single variable, you need two functions (one for above the x-axis and one for below it, in this case).ģ. A width dx, then, should given you a cross-section with volume, and you can integrate dx and still be able to compute the area for the cross-section. You know the cross-section is perpendicular to the x-axis. Integrate along the axis using the relevant bounds.Ī couple of hints for this particular problem:ġ. Find an expression for the area of the cross-section in terms of the base and/or the variable of integration.Ĥ. Find an expression in terms of that variable for the width of the base at a given point along the axis.ģ. Figure out which axis (and thus which variable) you'll be using for integration.Ģ. I won't give you the answer, but I'll offer a general strategy for questions of that variety:ġ.
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