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Use Simpson's Rule with $ n = 10 $ to approximate the area of the surface obtained by rotating the curve about the x-axis. Compare your answer with the value of the integral produced by a calculator.

$ y = \frac{1}{5} x^5 $ , $ 0 \le x \le 5 $

$$\approx 1,230,507$$

Applications of Integration

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Campbell University

University of Michigan - Ann Arbor

Idaho State University

Okay, we know that Ennis 10. And we know the interval with is Delta Axe is B minus over Andi, which is five. Minister Overton, which is 5/10 which is the same thing as 1/2. Therefore now what we have is we have our access and then we have acts to the fifth time's a square root of one plus acts to eight because remember, if we had why is 1/5 x to the fifth that we know are white primer? The derivative is next to the fourth. Therefore now we know Given Simpsons Rule again. We had N is 10. I stated this is the beginning of the problem indicating that we have 11 terms. Therefore, we have our two pi over five times Delta Acts, which we already establishes 1/2 divided by three time's up above zero plus four times affects one plus two times affects too plus four times of x three plus two times f of x four And then we have the second row weaken do X 53 x 10 So four times off of x five with two times off of XX plus four times and again. Remember, we had 11 terms. As I stated so from X zero all way to exit 10 and then, lastly, it's just one times f of x 10. So it's just the value because we know that multiplying by one just gives itself and then we're probably gonna want to use it calculated for this to end up with your value, and then we can round it like this. So, in other words, the solution is 1230507.2. This is what the calculator gives depending on how you round it and whether or not you put the decimals, it's the same answer.