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Volume of Revolution Calculator + Online Solver With Free Steps

A Volume of Revolution Calculator is a simple online tool that computes the volumes of usually revolved solids between curves, contours, constraints, and the rotational axis.

A function in the plane is rotated about a point in the plane to create a solid of revolution, a 3D object.

What Is a Volume of Revolution Calculator?

The Volume of Revolution Calculator is an online calculator that calculates an object’s volume as it rotates around a plane. However, the line must not cross that plane for this to occur.

When a function in the plane is rotated around a line in the plane, a solid of revolution is produced, which is a 3D object.

This solid’s volume can be determined via integration. The disc methodology, the shell approach, and the Centroid theorem are frequently used techniques for determining the volume.

For design, diagnostic imaging, and surface topography, volumes of revolution are helpful. Learning these solids is necessary for producing machine parts and Magnetic resonance imaging (MRI).

How To Use a Volume of Revolution Calculator?

You can use the  Volume of Revolution Calculator to get the results you want by carefully following the step-by-step instructions provided below. Follow the instructions to use the calculator correctly.

Step 1

Enter the expression for curves, axis, and its limits in the provided entry boxes.

Step 2

Press the “Calculate Volume” button to calculate the  Volume of the Revolution for the given data. It will also provide a detailed stepwise solution upon pressing the desired button.  

How Does a Volume of Revolution Calculator Work?

The Volume of Revolution Calculator works by determining the definite integral for the curves. In order to perform this kind of revolution around a vertical or horizontal line, there are three different techniques. These approaches are:

  • Disk Approach
  • Washer Technique
  • Shell Approach

Disk Method

The approach for estimating the amount of solid-state material that revolves around the axis is known as the disc method. Depending on the need, this could be along the x- or y-axis.

When the boundary of the planar region is coupled to the rotational axis, the disc approach is utilized. The disc method makes it simple to determine a solid’s volume around a line or its axis of rotation.

If we use the Disk technique Integration, we can turn the solid region we acquired from our function into a three-dimensional shape. 

We can determine the volume of each disc with a particular radius by dividing it into an endless number of discs of various radii and thicknesses.

Rotation Along X-axis

The volume of the solid of revolution is represented by an integral if the function to be revolved along the x-axis:

\[ V = \int_{a}^{b}(\pi R^2)(w) \]

Or,

\[  V= \int_{a}^{b}(\pi f(x)^2 )( \delta x) \]

\[  V= \int_{a}^{b}(\pi f(x)^2 )( dx) \]

Rotation Along Y-axis

The volume of the solid of revolution is represented by an integral if the function revolves along the y-axis:

\[ V = \int_{a}^{b} (\pi)   (R^2)(w) \]

Or,

\[  V= \int_{a}^{b}(\pi )  (f(y)^2 )( \delta y) \]

\[  V= \int_{a}^{b}(\pi   f(y)^2 ) ( dy) \]

Here, f(x) and f(y) display the radii of the solid, three-dimensional discs we constructed or the separation between a point on the curve and the axis of revolution.

Washer Method

The washer method is an integrating technique for calculating the volume of a solid in which the axis of rotation is taken perpendicular to the plane’s boundary rather than parallel to it.

Because we need to see the disc with a hole in it, or we can say there is a disc with a disc removed from its center, the washer method for determining volume is also known as the ring method.

We build a disc with a hole using the shape of the slice found in the washer technique graph. Thus, we deduct the inner circle’s area from the outer circle’s area.

The area will be determined as follows if R is the radius of the disk’s outer and inner halves, respectively:

\[ \pi \cdot R^2 \; – \; \pi r^2 \]

We will multiply the area by the disk’s thickness to obtain the volume of the function. Depending on the issue, both the x-axis and the y-axis will be used to determine the volume.

Rotation Along X-axis

The solid’s volume(V) is calculated by rotating the curve between functions  f(x) and g(x) on the interval [a,b] around the x-axis.

\[ V = \int_{a}^{b} \pi ([f(x)]^2−[g(x)]^2)(dx) \]

Rotation Along Y-axis

Similarly, the solid’s volume(V) is calculated by rotating the curve between f(x) and g(x) on an interval of [a,b] around the y-axis.

\[ V = \int_{a}^{b} \pi ([f(y)]^2−[g(y)]^2)(dy) \]

Here, f(x),g(x),f(y) and g(y) represent the outer radii and inner radii of the washer volume.

Shell Method

The shell method contrasts with the disc/washer approach in order to determine a solid’s volume. Cross-sectional areas of the solid are taken parallel to the axis of revolution when using the shell approach.

The area of a cylindrical shell with a radius of r and a height of h is equal to ‘2rh’. So, using the shell approach, the volume equals ‘2rh’ times the thickness. Any equation involving the shell method can be calculated using the volume by shell method calculator.

Solved Examples

Let’s explore some examples better to understand the working of the Volume of Revolution Calculator.

Example 1

Let’s calculate the solid’s volume after rotating the area beneath the graph of $ y = x^2 $ along the x-axis over the range [2,3].

Solution

Since the region’s edge is located on the x-axis. To obtain a solid region, the disc approach is utilized, and the graph of such a function is as follows:

The volume of a solid revolution using the disk method is calculated in the following manner:

\[ V= \int_{3}^{-2} \pi (x^2)^{2} dx \]

\[ V= \pi \int_{3}^{-2} \pi (x^4) dx \]

\[ V= \pi [ \frac{1}{5}  (x^5) ]^{3}_{-2} \]

\[ V= \pi [ \frac{243}{5} –  \frac{-32}{5}) ]  \]

\[ V=55 \pi \]

Example 2

Let’s calculate the solid’s volume by rotating the x-axis generated curve between $ y = x^2+2 $ and y = x+4.

Solution

y = x$^2$+2 

y = x+4

 x$^2$+2 = x+4 

 x$^2$ – x – 2 = 0 

(x+1)(x-2) = 0 

x = -1 and x = 2

The graph of the functions above will then meet at (-1,3) and (2,6), yielding the following result:

\[ V= \int_{2}^{-1} \pi [(x+4)^2−(x^2+2)^2]dx \]

\[ V= \int_{2}^{-1}  \pi [(x^2 + 8x + 16)−(x^4  + 4x^2 + 4)]dx \]

\[ V=\pi \int_{2}^{-1} (−x^4−3x^2+8x+12)dx \]

\[ V= \pi [− \frac{1}{5} x^5−x^3+4x^2+12x)] ^{2}_{-1} \]

\[ V= \pi [ \frac{128}{5} −(−\frac{34}{5})] \]

\[ V= \frac{162}{5} π \]

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