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(2 points) Consider the volume of the region shown below, which shows & right circular cone with top radius 1 cm and height 5 cm We have used the notation Dy fo...

Question

(2 points) Consider the volume of the region shown below, which shows & right circular cone with top radius 1 cm and height 5 cm We have used the notation Dy for AyWrite a Riemann sum for the volume; using the strip shown ad the variable y: Riemann sum = %Now write the integral that gives this volume: volume J where aand bFinally; calculate the exact volume of the region , using your integral volume (include units)

(2 points) Consider the volume of the region shown below, which shows & right circular cone with top radius 1 cm and height 5 cm We have used the notation Dy for Ay Write a Riemann sum for the volume; using the strip shown ad the variable y: Riemann sum = % Now write the integral that gives this volume: volume J where a and b Finally; calculate the exact volume of the region , using your integral volume (include units)



Answers

Use integration to find the volume of the following solids. In each case, choose a convenient coordinate system, find equations for the bounding surfaces, set up a triple integral, and evaluate the integral. Assume $a, b, c, r, R,$ and $h$ are positive constants. Cone Find the volume of a solid right circular cone with height $h$ and base radius $r$.

In question were given and tempering to co I'm doesn't to. One woman is 1 to 1 for minus square. The x d. Why? It was trying to draw the figure here you were. This will function inside. Here we read the ah rub a light on. It has the high acquittal for here. It's going down here. Then we have this ago on the place. Now and then, we have considered, uh, X is innocent. Accident. Why on dizzy? And we only considered xbg minus one and one. So the minus warm is here, and ones on one is near and minus one on the other side. And why was from 0 to 1, someone's here so it would draw on the vertical line up? Yeah. So this will be the region that we're looking for here. It would be like this, Andi. It would be on the other side. Yeah. So this will be the ricin. Will it fall? Okay, so this will be the bottom. We're looking from here, so I look at it. It's kind of hard to miss. Stayed here. No, in Italy. Very. This problem will do this department. The girl. So from 0 to 1. We came the outer for now, for the inner we have extremely the variable and yeah, not just dentition is from the month 1 to 1. Therefore, here inside this one is even. And this one is also the everything is even under forward. You have the two times now we will have the entire if don't wear have been recruited. Do that is the number minutes one joint have is there to one for minus X squared minus y squared the x no. And then we get equity the rooms to 12 on the entirely rift in this one Echo Jetta for X minus. Expect to have a Jew when it's quite square. X evil it and desert you one g y. Then we get the rooms or 212 And now it will be the one inside that actually getting good. You two for minus one of the Jew minus y square. And then do you? Why? And it wasn't if I've indicative rooms that you won four months 1/2 so good you It would be, uh, with him stew as well. So it minus one. Good. You seven minus two y square do you? Why? And then we will get equal to entirely riveted this one in calculus. Want seven Y minus two Weber 3/3. Even though its energy to one didn't get Nico Judah seven minus two and three. Then we get you got your 21 minus two. So you could you 1903.

Well, I problem number 79. So using similar triangles. If we extend the trust them to a cone, then the height is gonna be Rh over R minus are And therefore the volume of the festoons gonna be integral from sort H and the girl from zero ar minus Z times are minus are for H Then we integrate from 0 to 2 pi of a d thera t a d c then plugging in our function Medicals pi over three times are squared class R r plus R square tons each.

In description we have to find the volume bounded by what it will do squared of X. X equal to one. Why equal to zero and access X equal to one. So here we have to scare the solid opened by rotating each region around the indicated access, which is X equal to one. So here from the graph you can see that the volume of each yellow slides when the figure blow there already around X. Axis of revolution. So here you can see that Access axe equaled one. So here we know that X equal to weiss quiet so radius of task or equal to one Negative Wide Square. Now volume of slides approximate, will you? By the time I Squire time Delta Y. So we put the value of us. So here we get by time It's quite off. one negative wide Squire, delta Y. Now adding the world volume of solid, approximate value is sigma by time I Squire time delta Y. So here sigma by time one negative wide Squire to the power to Guy Delta Y. There is a way the thickness of each slice tends to zero to obtain a definite intrigue. Als since the curves intersect. Act, why can't do zeal? Hey Why equal to 1? So here volume of sorry equal to we equal to definite intrigue. All of by time one negative wide square to the power to dy from 0- one. So here we equal to why? Time definitely intrigued all of one negative two times y Squire. What to do? Why did the powerful B. Y. From 0 to 1? So now here we have to find anti anybody of 1 -2 times y squared plus why do the powerful is by time why negative two upon three like you. Plus Why to the power five upon 5 from 0 to 1. So here Upper value is one. So we put here. Why equal to one. So here we get weak. Will do by time one. Now you do two upon three. Father do one upon five time. Sorry, negative bye time. Zero, negative two times 2 upon three times 0 plus 0.5. So here you can see that we will do 85 upon 15. That is the value of the volume of solid. So it is our final answer.


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