In mathematics, engineering, and manufacturing, a solid of revolution is a solid figure obtained by rotating a plane curve around some straight line (the axis of revolution) that lies on the same plane.
Assuming that the curve does not cross the axis, the solid's volume is equal to the length of the circle described by the figure's centroid multiplied by the figure's area (Pappus's second centroid Theorem).
A representative disc is a three-dimensional volume element of a solid of revolution. The element is created by rotating a line segment (of length w) around some axis (located r units away), so that a cylindrical volume of πr2w units is enclosed.
Homework Statement
Hi. I'm asked to find the volume of the solid bounded by the paraboloid
4z=x^2 + y^2 and the plane z=4
I have drawn the graph in 3D but I'm unsure of how to set up the integral. Also, how does one decide to use double integrals/triple integrals when finding volume?
Homework Statement
Find the volume of the solid formed by rotating the region enclosed by the following equations about the Y-AXIS.
y= e^(3x)+5
y=0
x=0
x= 1/2
Homework Equations
The Attempt at a Solution
I keep getting the answer wrong. I broke the problem into two parts: solved a...
Homework Statement
Sketch the solid E bounded by the cylinder x = y^2 and the planes z = 3 and x + z = 1, and write down its analytic expression. Then, use a triple integral to find the volume of E.
The Attempt at a Solution
Was wondering if someone could have a go at drawing this sketch...
Homework Statement
f(x) =e^x and g(x)= ln(x)
Find the volume of the solid generated when the region enclosed by the graphs of f and g between x=1/2 and x=1 is revolved about the line y=4
Homework Equations
v= pi* integral( f(x)^2 - g(x)^2 dx)
The Attempt at a Solution
SO for...
Indicate the method you use to set up the integrals (do not integrate) that give the volume of the solid generated by rotating the region R around:
The region R is bounded by the curves y=x, x= 2-y^2 and y=0
i.) the x-axis
ii.) the y-axis
iii.) the line x= -2
iv.) the line y= 1
work...
Hello folks, I was wondering how to set up a volume of the solid of revolution about a line in the form of a line equation. if i wanted to find the volume about a line of x/4 would I simply find it as v=pi*integral (f(x/4)^2)dx or is there a method I'm missing all togeather?
Find the volume of the solid under the graph of z=sqrt(16-x^2-y^2) and above the circular region x^2+y^2<=4 in the xy plane
I know I must go to polar. So z=sqrt(16-r^2). Does r range from 0-2? I am not sure what theta ranges from (0-2pi)? I set up the integral as int(int r*sqrt(16-r^2)...
Hello ppl. I have a problem in finding out the volume of solid formed by the revolution of one loop of lemniscate of bernoulli ( r²=a²cos2θ) about the initial line θ =0
Using the relevant forumula for the volume of the solid generated by the revolution of one loop of the polar curve about the...
Homework Statement
Find the volume V of the solid bounded by the graph
x2+y2=9 and
y2+z2=9
Homework Equations
The Attempt at a Solution
When I started this problem, I thought it was a perfect sphere with the center points (0, 0, 0). And then I thought, "Why do I need calculus, it's...
Homework Statement
y = x³ y= 0 and x = 1
and its revolved around the line x = 2
okay i have drawn the graph of y = x³ and other paramaters, but when i get ther area being rotated it produces a hollow center. how do i go about finding the volume?
would it be a washers i don't...
Homework Statement
Find the volume of the solid generated by revolving the region bounded by the graph of
y = x3 and the line y = x,
between x = 0 and x = 1,
about the y-axis.
Homework Equations
\pi\overline{1}\int\underline{0}[R(x)^{2}-[r(x)]^{2}dx
The Attempt at a Solution...
Homework Statement
Find the volume of a solid formed by revolving the region bounded by graphs of:
y=x^3
y=1
and
x=2
Homework Equations
\pi0\int2(x^3)dx
The Attempt at a Solution
x^7/7 with boundaries of [0,2]
Am I on the right path?
Homework Statement
Find the volume of the solid of revolution:
F(x)=2x+3 on [0,1]
Revolved over the line x=3 and y=5
Homework Equations
Shell Method: 2\pi\int^{b}_{a}x[f(x)-g(x)]dx
obviously just sub y for dy
Disk Method: /pi/int^{b}_{a}[F(x)^{2}-G(x)^{2}dx
Homework Statement
Find the volume of the solid of revolution:
F(x)=2x+3 on [0,1]
Revolved over the line x=3 and y=5
Homework Equations
Shell Method: 2\pi\int^{b}_{a}x[f(x)-g(x)]dx
obviously just sub y for dy
Disk Method: /pi/int^{b}_{a}[F(x)^{2}-G(x)^{2}dx
The Attempt at a...
Homework Statement
find the volume of the solid generated by rotating the circle (x-10)^2 + y^2 = 36 about the y-axis
Homework Equations
disk method: \pi\int [R(x)]^2dx
shell method: 2\pi\int (x)(f(x))dx
The Attempt at a Solution
y = \sqrt{36-(x-10)^2}dx[\tex]
\\\pi\int...
find the volume of the solid resulting when the region enclosed by the curves is revolved around y-axis.
x=\sqrt{1+y} x=0 y=3
I am using this integral...
V=\int_{-1}^3[\pi(\sqrt{1+y})^2]dy
and I am getting the wrong answer.
I think it is just arithmetic, but are my bounds...
Ok, I'm supposed to found the volume of the solid that is created after rotating the line f(x) = 2x-1 around the x axis. The limits are y=0 x=3 and x=0. I've been trying for about and hour, and keep getting the answer: 46.0766. I've done the integration tons of times, splitting the problem...
so i have one problem and i just need to know if my integral is right. any help would be greatly appreciated
1. a ball of radius 10 has a round hole of radius 5 drilled through its center. find the volume of the resulting solid.
i know volume of cylinder removed is pi*5^2*10*sqrt(3)
because...
Hi
Could someone please give me an idea on how to go about this problem
Find the volume of the curve genereated by revolving the area between the curve y =(cos x)/x and the x-axis in the interval pie/6 to pie/2
Thanks a lot..
Here is the problem:
Find the volume of the solid that is bounded above by the cylinder z = 4 - x^2, on the sides by the cylinder x^2 + y^2 = 4, and below by the xy-plane.
Here is what I have:
\int_{-2}^{2}\int_{-\sqrt{4 - x^2}}^{\sqrt{4 - x^2}}\int_{0}^{4 - x^2}\;dz\;dy\;dx\;=\;12\pi...
"simple" shell
I know this is relatively simple, but I'm a little rusty. Could someone help me out? We want to find the volume of the solid obtained by rotating the region bounded by the curves y=x^4 and y=1 about the line y=7 using the cylindrical shell method.
According to my book the...
The base od a solid is a region in the 1st quadrant bounded by the x-axis, y-axis and the line x+2y=8. If cross sections of the solidperpendicular to the x-axis are semicircles, what is the volume of the solid?
How come the answer isn't just the intgegral from 0-8 of 1/2pi(4-x/2)^2
Hello,
I am still unsure of my ability to evaluate the volume of a solid using triple integrals. Here is my question:
Now I know that the intersection of the two paraboloids is
9 = x^2 + y^2.
But I am unsure how to set up the triple integral. I was thinking of splitting the volume...
Hello,
I am having trouble setting up triple integrals to find a volume of a given solid. Here is one of the questions with which I am having trouble.
Now I can see that the projection of the solid on the xy plane is the circle x^2 + y^2 = 9. And I think I can visualize the plane z = y +...
The quadrant of the ellipse \frac{x^2}{a^2}+\frac{y^2}{b^2} = 1. lying in the first quadrant, revolves about the line joining the extremities of the major and minor axis. Show that the volume of the solid generated is \frac{\pi a^2 b^2}{\sqrt{a^2+b^2}} (\frac{5}{3} - \frac{\pi}{2}).
I tried...