Find The Volume Of A Solid Calculator Desmos

Visualize with Desmos, Calculate Here

Calculate Volume (f(x) = cx^n)

Visualization Aid

Approximate view of f(x) and the region being rotated (not to exact scale for all inputs). Visualize interactively with Desmos!

What is Finding the Volume of a Solid of Revolution (and Desmos)?

Finding the volume of a solid of revolution is a concept in calculus used to determine the volume of a three-dimensional object formed by rotating a two-dimensional curve (or region) around an axis. Imagine taking an area under a curve f(x) between x=a and x=b and spinning it around the x-axis or y-axis; the shape you get is a solid of revolution. Our find the volume of a solid calculator desmos focuses on the Disk Method for functions like f(x)=cx^n rotated around the x-axis.

Desmos is a powerful online graphing calculator that is excellent for visualizing these concepts. You can graph the function, see the area, and imagine (or even animate with some tricks) the rotation. While this calculator performs the volume calculation, using Desmos alongside it can greatly enhance your understanding of how the solid is formed. This tool acts as a find the volume of a solid calculator desmos aid by calculating what Desmos helps you visualize.

This calculator is useful for students learning calculus, engineers, and anyone needing to find the volume of such solids for simple polynomial functions. A common misconception is that you always integrate f(x); however, for the Disk Method around the x-axis, you integrate π * [f(x)]².

Volume of Solid (Disk Method for f(x)=cx^n) Formula and Explanation

When we rotate the region under the curve y = f(x) = cx^n between x = a and x = b around the x-axis, we can imagine slicing the resulting solid into infinitesimally thin disks. Each disk has a radius f(x) and thickness dx. The volume of one disk is dV = π * [f(x)]² dx.

To find the total volume, we integrate this expression from a to b:

Volume (V) = ∫[a,b] π * [f(x)]² dx

For f(x) = cx^n, this becomes:

V = ∫[a,b] π * (cx^n)² dx = π * c² ∫[a,b] x^(2n) dx

The integral of x^(2n) is x^(2n+1) / (2n+1) (provided 2n+1 ≠ 0). So,

V = π * c² * [x^(2n+1) / (2n+1)] evaluated from a to b

V = π * c² * ( (b^(2n+1) / (2n+1)) – (a^(2n+1) / (2n+1)) )

This is the formula our find the volume of a solid calculator desmos tool uses.

Variables Used

Variable	Meaning	Unit	Typical Range
c	Coefficient of x^n	Dimensionless (or units to make f(x) have length units)	Any real number
n	Exponent of x	Dimensionless	Any real number (2n+1 ≠ 0)
a	Lower limit of integration	Length units	Any real number
b	Upper limit of integration	Length units	b ≥ a
V	Volume	(Length units)³	≥ 0

Practical Examples

Example 1: Volume of a Paraboloid

Let's find the volume of the solid generated by rotating y = x² (so c=1, n=2) from x=0 to x=2 around the x-axis.

• c = 1
• n = 2
• a = 0
• b = 2

Using the formula: V = π * 1² * ( (2^(2*2+1) / (2*2+1)) – (0^(2*2+1) / (2*2+1)) )

V = π * ( (2^5 / 5) – (0^5 / 5) ) = π * (32/5) ≈ 20.106 cubic units.

Our find the volume of a solid calculator desmos will give this result.

Example 2: Volume from y = √x

Find the volume of the solid formed by rotating y = √x (so c=1, n=0.5) from x=1 to x=4 around the x-axis.

• c = 1
• n = 0.5
• a = 1
• b = 4

2n+1 = 2*0.5+1 = 2

V = π * 1² * ( (4^(2) / 2) – (1^(2) / 2) ) = π * ( (16/2) – (1/2) ) = π * (15/2) = 7.5π ≈ 23.562 cubic units.

How to Use This find the volume of a solid calculator desmos

1. Enter the Coefficient (c): Input the value of 'c' from your function f(x) = cx^n.
2. Enter the Exponent (n): Input the value of 'n'. For √x, n is 0.5; for x³, n is 3.
3. Enter the Lower Limit (a): Input the starting x-value for the integration.
4. Enter the Upper Limit (b): Input the ending x-value. Ensure b is greater than or equal to a.
5. Calculate: The calculator automatically updates, or click "Calculate Volume".
6. Read the Results: The primary result is the volume. Intermediate values and the formula used are also shown.
7. Visualize: The canvas shows an approximation of the function and the area being rotated. For better dynamic visualization, use Desmos itself, then use our calculator for the volume number. You can find more on the disk method explained page.
8. Reset or Copy: Use the "Reset" button to go back to default values or "Copy Results" to copy the output.

This find the volume of a solid calculator desmos companion helps you get the numerical answer while you explore visually with Desmos.

Key Factors That Affect Volume Results

• The function f(x) (c and n): Larger values of |c| or n (for x>1) generally lead to larger f(x) values, thus a larger radius for the disks and greater volume.
• The Interval [a, b]: A wider interval (larger b-a) means more disks are being summed, usually resulting in a larger volume. The position of the interval also matters depending on f(x).
• Axis of Rotation: This calculator assumes rotation around the x-axis. Rotating around the y-axis or another line would require a different formula (like the Shell Method or Washer Method if there's a hole, see our washer method calculator).
• The Power (2n+1): If 2n+1 is zero (n=-0.5), the integral form changes. This calculator handles it by preventing n=-0.5.
• Whether b ≥ a: The upper limit b must be greater than or equal to the lower limit a for a standard volume calculation. If b < a, the result would be negative, representing volume with reversed integration direction.
• Units: If a and b are in meters, and f(x) represents meters, the volume will be in cubic meters. Be consistent with units.

Understanding these factors is crucial when using any find the volume of a solid calculator desmos or similar tool, including our calculus calculators.

Frequently Asked Questions (FAQ)

Q: What is the Disk Method? A: The Disk Method is used to find the volume of a solid of revolution when the region being rotated is flush against the axis of rotation. It involves summing the volumes of infinitesimally thin disks. Our find the volume of a solid calculator desmos uses this for f(x)=cx^n around the x-axis.

Q: How does this relate to Desmos? A: Desmos is excellent for graphing f(x), seeing the area between a and b, and visualizing the 3D solid that's formed. This calculator computes the volume of that solid for f(x)=cx^n, complementing Desmos's visual strength.

Q: Can this calculator handle any function f(x)? A: No, this specific calculator is designed for functions of the form f(x) = cx^n. For more complex functions, you'd need numerical integration or symbolic integration, which are beyond this tool's scope but can be set up in Desmos for visualization and approximation.

Q: What if I rotate around the y-axis? A: You would need to use the Disk Method with respect to y (if x is a function of y) or the Shell Method. This requires a different formula and setup. See our shell method tutorial.

Q: What if the region is not against the x-axis (Washer Method)? A: If you rotate a region between two curves, f(x) and g(x), you get a solid with a hole, and you'd use the Washer Method. See our washer method calculator.

Q: Why did I get a "2n+1 cannot be zero" error? A: If n = -0.5, then 2n+1 = 0, and the integration formula x^(2n+1)/(2n+1) involves division by zero. The integral of x^-1 is ln|x|, a different form. This calculator requires n ≠ -0.5.

Q: How accurate is the calculator? A: For functions f(x)=cx^n (where n ≠ -0.5), the calculator uses the exact analytical integral, so the result is very accurate, limited only by standard floating-point precision.

Q: Can I find the volume for x=a to x=b if a > b? A: Yes, but the result will be the negative of the volume from b to a. It's conventional to integrate from the smaller x-value to the larger x-value for positive volume. Our calculator allows b>=a.