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Find the coordinates of the centroids of the given figures. Each region is covered by a thin, flat plate. The solid generated by revolving the region in the first quadrant bounded by $y^{2}=4 x$ and $x=1$ about the $x$ -axis.

Calculus 2 / BC

Chapter 26

Applications of Integration

Section 4

Centroids

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Harvey Mudd College

Idaho State University

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So this curve why equals negative? X cube can be re written as X equals negative wide the 1/3. And, um, we need to know where the two curves intersect. And that happens at why equals one. So that will be our bounced. Integration abounds of integration. And also, of course, and zero, they are other about immigrations. We get 0 0 to 1 of, um pi r squared. So now, um, applied that big r squared minus little r squared. Um, And then, of course, the, uh, variable of integration is why, Cindy? Why? Um and big r is, um negative. No negative. Why? To the 1/3 um, minus negative to this, Plus to you and little R off the smaller radius is just three, you know, negative one. Minus negative too. Is our started. You know, it's negative, but three, because we only care about the length. So this integral is going to be the integral from zero 21 of pi times big R squared, minus little R squared, which is going to B. Why? To 2/3 power. Oh, miners too wide to the 1/3 power plus three integrated. Do you? Why and so this. It's going to be pie. We anti differentiate with plug in the numbers. PI times 3/5 minus two times 3/4 plus three That's our answer.

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If you write pi (to the first two decimal places of 3.14) backwards, in big, block letters it actually reads "PIE".

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