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If the infinite curve $ y = e^{-x} , x \ge 0 $, is rotated about the x-axis, find the area of the resulting surface.

$$

\pi[\sqrt{2}+\ln (1+\sqrt{2})]

$$

Applications of Integration

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Missouri State University

Campbell University

Harvey Mudd College

we have a number 28 28. Which question says that we have Girl, this is X. This is why we have a curb vital to areas to the power minus X. So car would look like this. It is to depart minus X. Okay, So it has given that if it is rotated about its success we have to find areas resulting surface if it will be rotated about X X is so it will be like this and we have to find the safe surface area of the resulting surface like area of this. So we have a formula for this that if we need to find if you need to find the area so area becomes equal to a to B to buy why DS were DS is equal to one place. Do you buy by DX whole square and two DX So let us find DS first. So d s will be equal to one. Plus we have y equals two years by minus six. So do you have any actual B minus series to power minus x whole square DX. So one place it is to the power minus two x dx. So this is the value of Diaz. So let us plug in and formula for area. Okay, So area will become equal to from a to B, which means we have to just to use from zero to infinity 02 infinity Because we have to take a from take the same curve when I get an ecology, Russell, these part will be when we did. We should be greater than equal to zero. So the limit on X it will be so. The limit it will be will be from zero to infinity. Okay, this is a limit on X because the curve is moving towards infinity and starting from Mexico to zero. Okay, No plug in the values value become to buy. And why? Why? Means it is to the power minus X and disease. One place it is to the power minus two x the X. So now let us plug in. It is to the power minus X equal to you because we have to substitute something we have to substitute. Had to use the method method of substitution minus areas of power minus x dx equal to do you. Okay, so it is about minus x d X will be equal to minors. Do you? Now? If we have X equal to zero, we have to just change the limit X equal to zero. So you will be equal to From here you will be equal to one. When we have X equal to infinity, you will be equal to zero because areas depart minus infinity will tends to zero will tend to zero. So our area will become equal to from, uh, when 20 minus. Do you? And we can take outside because it is constant. So with negative mind, sign one place you Esquire, Do you? Okay, so this is from 1 to 0. So if you need to change this from 0 to 1, this negative can be just use here. So this will be to buy one. Plus you Esquire, Do you? Now we have formula for this. That to buy 0 to 1 formula is you buy two one. Plus you Esquire under rule plus one by two. Ln you place one plus years square in the root. Okay, so this is two pi 0 to 1. If you plug a start plugging the limits. So let us plug in. You equal to one first everywhere. So it will be one by two under two, plus one by two. Ellen, one place under two. Yeah, And if you plug in zero So this will become zero, and this will become And this will also become zero. Because this will become Ellen one. And Ellen, one is always zero. So we have let us multiply with two pi. So he'll be having Bye. The route to less Ellen. One place route to Squire units. So this is a required area. Thank you so much.