Volume of solid Definition and 77 Threads

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.

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  1. P

    Volume of solid bounded by paraboloid and plane.

    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?
  2. A

    Volume of Solid: Find Y-Axis Rot. Region

    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...
  3. K

    Sketching and Calculating the Volume of Solid E

    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...
  4. N

    Finding Volume of Solid Revolved About Line y=4

    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...
  5. J

    Setting Up Integrals to Find Volume of Solid Rotated Around Region R

    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...
  6. J

    How to set up a volume of solid of revolution about a line other than the x axis

    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?
  7. D

    Volume of solid under graph and above circular region

    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)...
  8. R

    Volume of solid formed by revolution of one loop of Lemniscate of bernoulli

    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...
  9. L

    Solution to Finding Volume of Solid Using Calculus

    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...
  10. C

    What is the volume of a torus revolved around a line?

    volume of the torus, by revolving around line x = 3 inside circle x^2 + y^2 = 4 i got x = sqrt(y^2 - 4) and what would v = ?
  11. C

    Volume of Solid Generated by Revolving y = x³ around x = 2

    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...
  12. F

    Volume of solid revolving about y-axis

    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...
  13. F

    Calculating Volume of Solid Formed by Revolving Region

    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?
  14. D

    Finding Volume of Solid Revolved Around x=3, y=5

    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
  15. D

    Volume of Solid of Revolution: 115.19 and 77.206

    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...
  16. R

    Volume of solid and fluid force

    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...
  17. Saladsamurai

    Volume of Solid Revolved Around Y-Axis: Bounds Check

    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...
  18. C

    Finding Volume of Solid Rotated X-Axis

    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...
  19. N

    Solve Volume of Solid: y=x, y=x^1/2, y=1

    volume of solid:( hi..I tried to solve it but ı couldn't .book says the answer is pi /6...please help me. question is; y=x and y=x^1/2 about y =1
  20. B

    Is My Integral Calculation Correct for Finding the Volume of a Drilled Sphere?

    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...
  21. S

    Finding the Volume of a Revolved Curve: y = (cos x)/x from pi/6 to pi/2

    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..
  22. V

    Calculating Volume of Solid Bounded by Cylinders and Plane

    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...
  23. P

    Calculating Volume of Solid Using Cylindrical Shell Method

    "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...
  24. R

    Volume of Solid w/ Semicircular Cross Sections in 1st Quadrant

    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
  25. W

    Finding The Volume Of Solid Using Triple Integrals II

    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...
  26. W

    Finding The Volume Of Solid Using Triple Integrals

    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 +...
  27. himanshu121

    Volume of Solid Generated by Ellipse Quadrant Revolving About Major/Minor Axis

    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...
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