I need help on this singular question relating to finding the volume of a solid that is obtained. Any help is appreciated, please show work so I can learn better for next time. Thanks!
"Find the volume of the solid obtained by rotating the region bounded by y=x^2+1 and y=9-x^2
I too would like to know the answer to this question!
It is about the x axis, sorry for not including that!
first sketch graph
where do they cross ?
x^2 + 1 = 9 - x^2
2 x^2 = 8 that is handy !
x^2 = 4
x = 2
y = 5 cross at ( 2 , 5 )
so you want around x axis, integrate from x = 0 to x = 2
upper radius = 9-x^2
lower radius = x^2 + 1
area of outer circle = pi (9-x^2)^2
area of inner circle = pi (x^2 + 1)^2
wee want integral of difference times dx from x = 0 to x = 2
pi * int [ (81 - 18 x^2 + x^4) - (x^4 + 2 x^2 + 1) ] dx from 0 to 2
pi * int [ -20 x^2 + 80 ] dx from 0 to 2
pi [ -20/3 x^3 + 80 x] at 2
pi ( -53 1/3 + 160)
pi (106.66)
about 335
When I tried this, I got exactly double what you got. Why is that?
I think your integral needs to be from -2 to 2, not 0 to 2
sorry, you are right.
Sure! To find the volume of the solid obtained by rotating the region bounded by the curves y = x^2 + 1 and y = 9 - x^2, we'll use the method of cylindrical shells. Here's a step-by-step explanation:
Step 1: Find the points of intersection.
To find the points of intersection between the two curves, set them equal to each other:
x^2 + 1 = 9 - x^2
Simplifying the equation, we have:
2x^2 = 8
x^2 = 4
x = ± 2
So, the points of intersection are (-2, 5) and (2, 5).
Step 2: Determine the height of the shell.
The height of each cylindrical shell is given by the difference between the y-values of the curves at a particular x-value. In this case, the height of the shell at any x-value is y = (9 - x^2) - (x^2 + 1), or y = 8 - 2x^2.
Step 3: Determine the radius of the shell.
The radius of each cylindrical shell is given by the x-value of the curve being rotated. In this case, the radius is simply x.
Step 4: Setup and integrate the formula for volume.
The formula for the volume of a cylindrical shell is V = 2πrh, where r is the radius and h is the height.
The total volume of the solid can be found by integrating this formula over the x-interval from -2 to 2:
V = ∫[a to b] 2πrh dx = ∫[-2 to 2] 2π(x)(8 - 2x^2) dx
Now, integrate the equation:
V = 2π ∫[-2 to 2] (8x - 2x^3) dx
V = 2π [4x^2 - (1/2)x^4] |[-2 to 2]
Step 5: Evaluate the definite integral.
Plug in the values of x and evaluate the integral:
V = 2π [(4(2)^2 - (1/2)(2)^4) - (4(-2)^2 - (1/2)(-2)^4)]
V = 2π [(16 - 8) - (16 - 8)]
V = 2π [8 - 8]
V = 0
Hence, the volume of the solid obtained by rotating the region bounded by y = x^2 + 1 and y = 9 - x^2 is 0.
I hope this explanation helps you understand the process! Let me know if you have any further questions.