Question
Figure shows the graph ofy = x2 andy = 2-x2 Deterine the volume of the solid of revolution when the bounded region revolves 1808 about the Y-axisy = 22y =2-2Figure 3
Figure shows the graph ofy = x2 andy = 2-x2 Deterine the volume of the solid of revolution when the bounded region revolves 1808 about the Y-axis y = 22 y =2-2 Figure 3
Answers
Volume of a Solid of Revolution Find the volume of the solid of revolution generated by revolving the region bounded by the graph of $y=\frac{x}{x^{2}-4}$ and the $x$ -axis from $x=3$ to $x=5$ about the $x$ -axis, as shown in the figure below.
In this question we have to find the volume of the solid of revolution generated by evolving the reason bounded by the crop off by equals two for cyan X and the axis gone And the excesses from x equals two doodle two, x equals two by by two. About the by access. The volume is given by the integral by X times Joseon X. The X limit 02 by by two. Now we will use integration by parts In order to solve this integral. So like U equals two To buy X. on differentiating it it becomes the U equals two to buy the X. Similarly like T V equals two, we'll sign X. E. X. Or integrating both sides. It becomes equals to cynics. Now we will apply the formula for integration by parts. So we can buy at the integral S to buy X multiplied by the cosine X. The X limit zero by by two equals two to buy X times phoenix limit. They don't have a right to minus integral off my next times Dubai E X. Let me do you do. Bye bye to. Mhm. Now we will apply the limits so it becomes 25 times by by two. Multiplied by signed by by two minus 2. 5 times zero Multiplied by Science zero minus Dubai times integral of sine X. The X limit 0-5 or two on simply find it. We get by square minus 25 times minus cosine X Limit 0-5 x two. As the integral of this part is given by minus course I next value of signed by by police one And value signs you is zero. Now we will apply the limit so it becomes by square minus Dubai times Minour co signed by by two minus bracket minus for Science zero one, simplifying it bigger than answer which is by square minus. Mhm. Goodbye. Thank you.
So we are given why is equal to one over square look affects and the area is bounded by. Why is equal to zero. X is equal to two and excessive to six, and we are going to protect this area about via about the exact so use equal to in a cruel from 2 to 65 times Y squared DX. You can put the pie in the front so we can say musical comply kind integral from 2 to 6 and Y squared is one over x DX. The integral a one over X T X is equal to natural law effects of these equal to five times the natural law defects, with the limits off any creation going from 2 to 6. So these equal to pi times the natural log off six minus the natural off to on that is equal to read is equal to five times the natural off. Six divided by two, which would be threes or final answer. Volume is equal to five times the natural off. Three
So we are given The curved by is equal to equally power X and boundaries are why is equal to zero. X is equal to zero. X is equal to tooth. So we will people to the integral from 0 to 2 high times y skway, DX or easy could do in the grill from 0 to 2 and fortified in front of the grill and then eat the power X squared DX so that we is now equal to high times the integral from 0 to 2. Eat power, uh, two x dx and that would be equal to B is equal to five times e to the power to x over to limits of integration from 0 to 2. So we now have me is equal to pi kinds he to the power for over two minus e to the power zero over to 18 hours. Zero is one. So we have these equal to high over two and factor the one or two and Mackey's We would he to de power for minus one, and that is approximately equal to 84.1 nine to
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