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Find the exact area cf the surface obtained bY rotating the curve bout the X-axis Y =x3_ <X<...

Question

Find the exact area cf the surface obtained bY rotating the curve bout the X-axis Y =x3_ <X<

Find the exact area cf the surface obtained bY rotating the curve bout the X-axis Y =x3_ <X<



Answers

Find the exact area of the surface obtained by rotating the curve about the x-axis.

$ y = x^3 $ , $ 0 \le x \le 2 $

This question asked us to find exact area of the surface by rotating the curb about the X axis. As it stated, What we know we need to do is we know we need to first off, figure out the bounds. They've actually given this to us of exes between zero and three. This means zero and three are about. Now we know we pull out to pie because we know we're gonna be rotating it as it's stated in the problem. And then we know that we have Why squared is X plus one. What we know this means is that if axes y squared minus one, then D axe over D. Y is equivalent to why this is critical here. Because now what we know we have is remember when we're writing this were essentially plugging in using X. So that is the same thing. That's why you can consider this to be. And then we know that we need to integrate. So in order to do that, we know we're going to be We know we're going to be using the for the multiplication pattern. Essentially distribution eight times people see, is a B plus a C So it's from 0 to 3 to pie X plus five over four de axe. We know we're going to be integrating this. We use the power rule of increasing the exploded by one dividing by the new exponents. That's how you integrate. And then we plug in and what we end up with is pi. Over six time 17 squared of 17 minus five squared of five.

We no formal. We're gonna be using his intro from A to B to par. Why d us know. What we know is that our squared of one plus you to the axe is because it is gonna become one plus e to the X over to because we don't want the square root or more plugging into the formula and makes it a lot more complicated to read. So one plus either the axe over too Times D Backs member. It's important that we're following this formula over here. This is the same thing as two part comes expose you to the axe divided by two. Remember in Vieques over to taking the Inderal? It's still eating the X over two. He's kind of a special case. If you don't remember all the properties of the I would recommend reading up on it. But it's a special case that you should have learned about in a previous math class. Plugging in to part times one pussy over to minus 1/2 gives us pi times one plus e. And then this is the exact answer so we can leave this us

The first thing we should do is we should differentiate with respect to why So if we have four why we square this we end up with 16. Why squared? Which means we now have integral from wonder, too, to pi times. Why Times square off one plus 16 y squared Do you want now a little bit of U substitution If you was one for 16 where I squared What's under the square root? Then D'You is gonna be 32. Why di roi, which means the limits of integration. Now change. We now have power over 16 on the outside and we have 17 to 65 squirt of u d y. Okay, time to integrate. When we integrate, we know we're gonna be using the power rule, which means we increase the exponents by one, and then we divide by the new extra plug it in are two bounds have been given and we know this could be written is our solution because they've said exact area so we can leave our exact area as this

This question asked us to find the exact area of the serfs obtained by routine the curve about the X axis. Have they stated we're looking at between 1/2 of one, So those are bounds and because we're rotating it, we know we need to add to pie. Not we know. Our original is X cubed over six plus one over to X. This is the original expression we've been given. We know this is also gonna be multiplied by why Prime squared. So one plus 1/4 ext the fourth, minus one house plus one over four ox the fourth and again, As I stated, this is the formula listed in the textbook where you've got, like, two pi when they've got why prime squared to two pi r squared or whatever the formula is how you consider it to be. We know now we take the integral. We do this by increasing the exponent by one and then dividing by the new exponents. You can also simple for the fractions as you go along. If that makes it easier for you to understand and remember our bouncer from 1/2 to 1 and this was given in the problem. Now that we have this, we know you need to plug end. So we're plugging in upper minus lower and you're probably gonna need a calculator for this. We end up with approximately 1.3 pie or 3.227


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