We're learning disks, shells, and cylinders in school but we have a substitute and I've been trying to teach this to myself. Can you check them please? =) Thank you!
1) Find the volume of the solid formed when the region bounded by curves y=x^3 + 1, x= 1, and y=0 is rotated about the x-axis.
My answer is 23/14.
2) Find the volume of the solid of revolution obtained by revolving the region bounded by y=1/x and the lines x=pi/8 and x=pi/2 around the x-axis.
I got 6/pi for this one.
3) Find the area bounded by the curves f(x)= x^3 + x^2 and g(x)= 2x^2 + 2x.
My answer: 5/12
Can you tell me if I did them wrong, and if I did do them wrong what I Did wrong?
standard way to form volume if spun around x-axis
y = x^3 + 1 ---> this becomes the radius of your disk
we need the x-intercept,
let x^3 + 1 = 0
x = -1 ---> this becomes our left boundary
Volume = π∫(x^3 + 1)^2 dx from x = -1 to x = 1
= π∫(x^6 + 2x^3 + 1) dx
= π [ x^7/7 + x^4/2 + x] from -1 to 1
= π( 1/7 + 1/2 + 1 - ( -1/7) + 1/2 - 1)
= P π( 2/7 + 2)= 16π/7
for #2,
y = 1/x
so y^2 = 1/x^2
V = π∫1/x^2 dx from π/8 to π/2
= π [ - 1/x ] from π/8 to π/2
= π ( -2/π - (-8/π)
= -2+8 = 6
#3, first you need the intersection:
x^3 + x^2 = 2x^2 + 2x
x^3 - x^2 - 2x = 0
x(x^2 - x - 2) = 0
x(x-2)(x+1) = 0
they intersect at x = -1, x=0 and x=2
So you must find the volume separately from
x = -1 to x = 0 , and then from x = 0 to x = 2
from -1 to 0 , the cubic is the upper curve, while from
0 to 2, the parabola is the upper curve
So you have
V = π∫( (x^3 + x^2)^2 - (2x^2 + 2x)^2 ) dx from -1 to 0 + π∫( (2x^2 + 2x)^2 - (x^3 + x^2)^2 ) dx from 0 to 2
= etc.
Thank you very much Reiny. I understand it a bit better now.
I did it and I get 4653pi/105 but this is wrong. Have I calculated incorrectly?
Sure, I can help you check your answers!
1) To find the volume of the solid, formed when the region bounded by curves y = x^3 + 1, x = 1, and y = 0 is rotated about the x-axis, we can use the method of disks or washers.
Using the method of disks, we can approximate the volume by dividing the region into small disks with thickness Δx. The radius of each disk is given by the function y = x^3 + 1. The volume of each disk is then π * (radius)^2 * Δx.
Integrating this expression over the interval [0, 1], we get:
V = ∫[0,1] π * ((x^3 + 1)^2) dx
Evaluating this integral, we find:
V = π * ∫[0,1] (x^6 + 2x^3 + 1) dx
= π * [(1/7)x^7 + (1/2)x^4 + x] | from 0 to 1
= π * [(1/7) + (1/2) + 1]
= π * (23/14)
= 23π/14
So the correct answer is 23π/14, not 23/14.
2) To find the volume of the solid of revolution obtained by revolving the region bounded by y = 1/x, x = π/8, and x = π/2 around the x-axis, we again can use the method of disks or washers.
Using the method of washers, we can approximate the volume by dividing the region into thin washers with thickness Δx. The outer radius of each washer is given by the function y = 1/x, and the inner radius is zero (since we are revolving around the x-axis).
The volume of each washer is then π * (outer radius)^2 * Δx - π * (inner radius)^2 * Δx = π * (1/x)^2 * Δx.
Integrating this expression over the interval [π/8, π/2], we get:
V = ∫[π/8,π/2] π * (1/x)^2 * dx
Evaluating this integral, we find:
V = π * ∫[π/8,π/2] (1/x^2) dx
= π * [-1/x] | from π/8 to π/2
= π * [-1/(π/2) + 1/(π/8)]
= π * [-2/π + 8/π]
= π * (6/π)
= 6
So the correct answer is 6, not 6/π.
3) To find the area bounded by the curves f(x) = x^3 + x^2 and g(x) = 2x^2 + 2x, we need to find the points of intersection between these two curves.
Setting f(x) equal to g(x), we have:
x^3 + x^2 = 2x^2 + 2x
Rearranging and simplifying, we get:
x^3 - x^2 - 2x = 0
Factoring out an x, we have:
x(x^2 - x - 2) = 0
(x - 2)(x + 1) = 0
So the curves f(x) and g(x) intersect at x = 2 and x = -1.
To find the area between the curves, we integrate the difference between f(x) and g(x) over the interval [-1, 2]:
A = ∫[-1,2] (f(x) - g(x)) dx
Evaluating this integral, we find:
A = ∫[-1,2] (x^3 + x^2 - (2x^2 + 2x)) dx
= ∫[-1,2] (x^3 - x^2 - 2x^2 - 2x) dx
= ∫[-1,2] (x^3 - 3x^2 - 2x) dx
= (1/4) x^4 - x^3 - (x^2) | from -1 to 2
= (1/4) * 16 - 8 - 4 - (1/4) + 1 - 1
= 4 - 8 - 4 - 1 + 1 - 1
= - 8
So the correct answer is -8, not 5/12.
I hope this helps! Let me know if you have any further questions.