29 y x 2 y 0 x 0 x 3 find the volume of the solid 15763

Name: 29 y x 2 y 0 x 0 x 3 find the volume of the solid 15763
Uploaded: 2022-03-21T09:27:04-08:00
Duration: 6 min 49 s
Channel: Raman S
Description: 29 y x 2 y 0 x 0 x 3 find the volume of the solid 15763

Transcript

00:01 Hello students in this given question we have been given a curve that is x equal to so the given is so a curve x is equal to 2 square root of y and the initial conditions that x equal to 0 and so we have to find the volume of solid by rotating this curve across x equal to 0 and y equal to 9 and we have to find the volume of solid solid by rotating the curve about x equal to 0 so here this curve is or this line is x equal to 0 and this dotted line is y equal to 9 so here we have to rotate this above x equal to zero so to solve this problem so the given is x is equal to 2 root y and if we square both sides so we will get x square is equal to 4 from here we can write y is equal to x squared by four also this line so this curve this curve is x equal to root to root y and this line both intersect at this point so we can write here y is equal to nine and y is equal to x squared by four so we can write x square by four is equal to nine from here we can write x is equal to square root of 36 that is six so this point is 6 and 9 right now take a vertical element take a vertical element at distance of x that is we will take this vertical element at a distance of x whose thickness is dx as shown in the figure so when this cylinder or when this element is rotated about y x that is x equal to 0 it will form it will form a cylindrical so the height of cylindrical cell at an instant will be y minus the total height is 9 minus y so we can write this as 9 minus y we have calculated like we can write y is equal to x squared divided by 4 for and the radius of cylinder radius of cylinder that is r equal to x and the thickness is that is t is to d x so we can write volume of volume of cylindrical cell is equal to 2 pi r h t here r r is here r is the h is 9 minus x squared by 4 and r is x and t is d x so we can write this as 2 .9 x minus x q by 4 d x since we have to find the total volume cell and total volume of cylinder so the total volume of cylinder total volume of cylinder for this what we will have to do we will have to integrate this so the total volume will be integration of 2 .5, 9x minus x cubed divided by 4 dx.

05:04 And x is varying from 0 to 6.

05:07 So the limitability is 0 to 6.

05:09 And 2 pi is constant.

05:10 So we will take it out.

05:11 And here it will be 9x minus x cube divided by 4 dx from 0 to 6.

05:20 So if we integrate this, we will get 9x squared by 2 minus x to the power 4 divided by 4 multiplied by 4.

05:32 And the limit is from 0 to 6.

05:34 We have used the concept of like integration of x to the power n dx is equal to x to the power n plus 1 divided by n plus 1...