![]() ![]() To help us visualize this shape here, I've kind of drawn a picture Now obviously that changesĪs we change our X value. It's a right triangle, and then this distance this distance between that point and this point is the same as the distance between F of X and G of X. Isosceles right triangle sits along the base. Isosceles right triangle with a hypotenuse of the If you were to actually flatten it out, the cross sections would look like this. ![]() This cross section is going to look like this, if you were going to flatten it out. 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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