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And x equals 3 and x equals 1 are clearly the zeroes of this function right over here.And I want to take this region and rotate it around the y-axis. And the height right over here is going to be the value of my function. In this case, f of x is x minus 3 squared times x minus 1. Well, the radius right over here is just the horizontal distance between the y-axis and that x. So the circumference, in this case, is just going to be 2 pi times x. And so what is going to be the surface area of the outside?I've got the function y is equal to x minus 3 squared times x minus 1.And what I want to do is think about rotating the part of this function that sits right over here between x is equal to 1 and x equals 3.As an example Pinnacle Point's caves, in the southern coast of South Africa, provided evidence that marine resources (shellfish) have been regularly exploited by humans as of 170,000 years ago.
This usually requires what is commonly known as a "dating method".It is commonly assumed that if the remains or elements to be dated are older than the human species, the disciplines which study them are sciences such geology or paleontology, among some others.Nevertheless, the range of time within archaeological dating can be enormous compared to the average lifespan of a singular human being.Well, the outside surface area is just going to be the circumference times the height. And in this situation, in the situation we're looking at right over here, that's going to be 2 pi x times x minus 3 squared times x minus 1. So the volume of the shell-- shell volume-- is just going to be all this business times dx. And so now we're ready to integrate over the interval.So the volume of our entire shape is going to be the definite integral. And on the inside, we have x times f of x, which in our situation, is this business. Using the shell method, we have set up our definite integral for the volume of this strange-looking shape right over there.