find the volume of the region?

Question

Find the volume of the region enclosed by y=1+x^(.5), y=1, x=4, is rotated around the y-axis.

Answer 1

Use washer method.
v
= 16 pi(2) - int[pi(y-1)^4dy, y from 1 to 3
= (128/5)pi
----------
Use shell method.
v
= int[2pi x(1+sqrt(x)-1) dx], x from 0 to 4
= (128/5)pi

Answer 2

The region is three sided with the y=1,and x=4 lines being two of the sides, and the third side formed by the sideways parabola y = 1 + sqrt(x).

The area of the disks formed by points of the parabola rotated around the x-axis is pi * (1 + x^.5)^2 = pi(x + 2x^.5 + 1)

The area of the disks formed by points of the y-1line rotated is pi.

So the area of the washer formed by rotating the region is pi(x + x^0.5)

Integrating we get

pi ( 1/2 x^2 + 2/3 x^1.5)

evaluating from 0 to 4, we get

volume = pi(8 + 16/3) - pi(0)
= 40/3 pi

Answer 3

sahsjing has your answer.

An alternative way to verify (instead of using 2 different methods to compute the volume) is to compute the "cone" volume within your requested volume. The cone is
(using thin shells abt the y-axis)

integ from low limit x = 0 to x=4
2pi (x) [3 - (1+sqrtx)] dx = (2^5)pi / 5

Add this cone volume to sahsjing's computed volume
(2^7)pi / 5) and you should get the simple solid volume (2 by 4 rectangle abt y-axis): 2^5 pi (You do.)

Answer 4

you recognize the quantity of a cube, and the quantity of a sphere, or you are able to look them up, or derive them via calculus. the biggest link right this is to confirm the size of the sphere via the size of a cube. A cube has all aspects of equivalent length, and since the sphere is inscribed interior the cube, you recognize that the size of the cube's factor is comparable to the diameter of the sphere. Relate the diameter to the radius, and you are able to now confirm the quantity of the sphere besides because of the fact the cube. Now, you will desire to flow with the greater suitable of the two, that's the cube because of the fact the sphere is interior. Subtract the smaller quantity from the greater suitable quantity and you gets the the rest quantity. Edit: i'm surprised at how Dulara controlled to infer your cube is measured in centimeters once you in no way unique gadgets of degree.

Answer 5

Vol = pi x integral(lim1 , lim4) y^2 dx = approx 91pi/6