(40 pts total) You have a solid obtained by rotating the...

Name: (40 pts total) You have a solid obtained by rotating the (finite) region between curves f(x) = √(x) and g(x) = x^2 (on [0, 1]) about the y-axis. Find the volume of the solid obtained by rotating this region two ways.
Uploaded: 2023-07-23T19:12:36-08:00
Duration: 4 min 56 s
Channel: Sri K
Description: (40 pts total) You have a solid obtained by rotating the (finite) region between curves f(x) = √(x) and g(x) = x^2 (on [0, 1]) about the y-axis. Find the volume of the solid obtained by rotating this region two ways.

(40 pts total) You have a solid obtained by rotating the (finite) region between curves \(f(x) = \sqrt{x}\) and \(g(x) = x^2\) (on \([0, 1]\)) about the y-axis. Find the volume of the solid obtained by rotating this region two ways.

Transcript

00:01 Given that y is equal to x square and y is equal to twice x.

00:07 These two quantities are given as and we have to use the shell model to find the volume of a solid obtained by rotating the region bounded by these two about the x axis in the first part and in the second part we have to calculate this along the y axis.

00:27 So for this first we can find here x for the x axis.

00:32 So y is equal to x square and y is equal to 2.

00:35 X gives you x square it is equal to twice x.

00:38 Therefore x square minus twice x will be equal to 0.

00:43 That means if we take this x common so it will be x minus 2 it is equal to 0 which implies that x it is equal to 0 and x it is equal to 2.

00:54 This is the first and for the y axis we will take the y by 2 this x we have to equate so we will get y by 2 it is equal to under root of y.

01:08 So this will gives you y square by 4 it is equal to y.

01:13 Therefore y square will be equal to 4y.

01:16 Therefore y square minus 4y it is equal to 0.

01:20 So y into y minus 4 it is equal to 0.

01:25 This will gives you y it is equal to 0 and y it is equal to 4.

01:30 This are the integration limits for the a part that is for calculating along the x axis and these are for along calculating the y axis.

01:42 So we will calculate the a part first.

01:45 So for the a part the points of intersection are x it is equal to 0 and x it is equal to 2.

01:52 Now to determine the bounds of the integration the bounds of integration will be a will be equal to 0 and b it is equal to 0.

02:00 Since those are the x values where the curves intersect.

02:04 Now we set up the integral using the shell method formula.

02:08 Here we have to use the shell method formula.

02:11 We are using this because in the question we are given that we have to use this method only...

Question

(40 pts total) You have a solid obtained by rotating the (finite) region between curves \(f(x) = \sqrt{x}\) and \(g(x) = x^2\) (on \([0, 1]\)) about the y-axis. Find the volume of the solid obtained by rotating this region two ways.

Please give Ace some feedback