Find The Volume Rotated About Y Axis Calculator

Volume Calculator for x = k*y^n

This calculator finds the volume of the solid generated by rotating the region bounded by x = k*y^n, x=0, y=c, and y=d around the y-axis.

What is a Find the Volume Rotated About y-axis Calculator?

A "find the volume rotated about y-axis calculator" is a tool used to determine the volume of a three-dimensional solid formed by revolving a two-dimensional region around the y-axis. This process is a fundamental concept in integral calculus, specifically in finding volumes of solids of revolution. The calculator typically employs methods like the Disk Method, Washer Method, or Cylindrical Shells Method (when functions are given as y=f(x)) to compute the volume by integrating cross-sectional areas or shell volumes.

This particular calculator focuses on rotating a region bounded by a curve defined by x = k*yⁿ, the y-axis (x=0), and horizontal lines y=c and y=d around the y-axis, using the Disk Method.

Who should use it?

• Calculus students learning about volumes of revolution.
• Engineers and designers calculating volumes of symmetrical objects.
• Mathematicians and physicists working with integration applications.

Common Misconceptions

A common misconception is that rotating around the y-axis is always done using functions of y (x=f(y)). While the Disk/Washer method for y-axis rotation uses x=f(y), the Shell Method allows rotation around the y-axis using functions of x (y=f(x)). Our calculator uses x=f(y). Another is confusing the formulas for rotation about the x-axis and y-axis. Always check which axis is being used for rotation. The find the volume rotated about y axis calculator helps clarify these by focusing on y-axis rotation.

Find the Volume Rotated About y-axis Formula and Mathematical Explanation

To find the volume of a solid generated by rotating a region bounded by x=f(y), x=g(y) (with f(y) ≥ g(y)), y=c, and y=d around the y-axis, we use the Washer Method (a generalization of the Disk Method):

V = π ∫_c^d [(Outer Radius)² – (Inner Radius)²] dy

Here, the outer radius is f(y) and the inner radius is g(y) if we rotate the area between x=f(y) and x=g(y). If we rotate the area between x=f(y) and the y-axis (x=0), and assuming f(y) ≥ 0, the formula simplifies to the Disk Method:

V = π ∫_c^d [f(y)]² dy

Our find the volume rotated about y axis calculator deals with the specific case where f(y) = k*yⁿ and the region is bounded by x = k*yⁿ, x=0 (y-axis), y=c, and y=d. So, the volume is:

V = π ∫_c^d (k*yⁿ)² dy = π k² ∫_c^d y²ⁿ dy

The integral of y²ⁿ dy is y⁽²ⁿ⁺¹⁾/(2n+1), provided 2n+1 ≠ 0 (i.e., n ≠ -0.5). If n = -0.5, the integral is ln|y|.

So, if n ≠ -0.5: V = π k² [y⁽²ⁿ⁺¹⁾ / (2n+1)]_c^d = π k² (d⁽²ⁿ⁺¹⁾ – c⁽²ⁿ⁺¹⁾) / (2n+1)

If n = -0.5 (and c, d > 0): V = π k² [ln|y|]_c^d = π k² (ln|d| – ln|c|)

Variables Table

Variable	Meaning	Unit	Typical Range
k	Coefficient in x=ky^n	Depends on units of x and y	Any real number
n	Exponent in x=ky^n	Dimensionless	Any real number
c	Lower limit of integration (y-value)	Units of y	Any real number, d ≥ c
d	Upper limit of integration (y-value)	Units of y	Any real number, d ≥ c
V	Volume of the solid of revolution	Cubic units	Non-negative

Variables used in the volume calculation for x=ky^n rotated about y-axis.

Practical Examples (Real-World Use Cases)

Example 1: Volume of a Paraboloid-like Shape

Suppose we want to find the volume of a solid generated by rotating the region bounded by x = y², x=0, y=0, and y=2 around the y-axis. Here, k=1, n=2, c=0, d=2.

Using the find the volume rotated about y axis calculator or the formula:

2n+1 = 2(2)+1 = 5

V = π * 1² * (2⁵ – 0⁵) / 5 = π * (32 – 0) / 5 = 32π / 5 ≈ 20.11 cubic units.

Example 2: Volume with n = 0.5

Consider the region bounded by x = 2√y (x = 2y^0.5), x=0, y=1, and y=4, rotated about the y-axis. Here, k=2, n=0.5, c=1, d=4.

2n+1 = 2(0.5)+1 = 2

V = π * 2² * (4² – 1²) / 2 = 4π * (16 – 1) / 2 = 4π * 15 / 2 = 30π ≈ 94.25 cubic units.

You can verify these with the find the volume rotated about y axis calculator.

How to Use This Find the Volume Rotated About y-axis Calculator

Using our find the volume rotated about y axis calculator is straightforward:

1. Enter the Coefficient (k): Input the value of 'k' from the equation x = k*yⁿ.
2. Enter the Exponent (n): Input the value of 'n' from the equation x = k*yⁿ.
3. Enter the Lower Limit (c): Input the starting y-value for the integration.
4. Enter the Upper Limit (d): Input the ending y-value for the integration (ensure d ≥ c).
5. Calculate: The calculator automatically updates the Volume and intermediate results as you type. You can also click "Calculate Volume".
6. Read Results: The "Primary Result" shows the calculated volume. Intermediate values and the formula used are also displayed.
7. Reset: Click "Reset" to return to default values.
8. Copy: Click "Copy Results" to copy the main result and intermediate values to your clipboard.

The graph dynamically updates to show the curve x=ky^n between y=c and y=d.

Key Factors That Affect Volume Results

• The Function x=f(y): The shape of the curve being rotated (defined by k and n in our calculator) directly determines the radius of the disks or washers, significantly impacting the volume. A larger 'k' or 'n' (for y>1) generally increases the radius and thus the volume.
• Limits of Integration (c and d): The interval [c, d] along the y-axis defines the height or extent of the solid of revolution. A larger interval (greater d-c) typically results in a larger volume.
• The Axis of Rotation: We are rotating about the y-axis. Rotating the same region about the x-axis would yield a different solid and volume (and require a different formula/method, likely Shells if starting with x=f(y)).
• Value of n being -0.5: If n = -0.5, the integration involves a natural logarithm, leading to a different volume calculation compared to other 'n' values. The calculator handles this special case.
• Whether c and d are positive when n=-0.5: If n=-0.5, the function is x=k/√y, and ln|y| is involved. We require c, d > 0 for ln|y| to be real.
• The square of k: The volume is proportional to k², so doubling k quadruples the volume, keeping other factors constant.

This find the volume rotated about y axis calculator is specifically for rotating the area between x=ky^n and x=0 around the y-axis.

Frequently Asked Questions (FAQ)

What if my function is not in the form x = k*y^n?

This specific find the volume rotated about y axis calculator is designed for x=ky^n. For other functions x=f(y), you would need to calculate π ∫[f(y)]² dy manually or use a more general symbolic integrator.

What if the region is bounded by two curves, x=f(y) and x=g(y)?

You would use the Washer Method: V = π ∫[(f(y))² – (g(y))²] dy, assuming f(y) is the outer radius. Our calculator handles f(y)=ky^n and g(y)=0.

How do I find the volume if I rotate around the x-axis?

If you have x=f(y) and rotate around the x-axis, you'd typically use the Cylindrical Shells method with respect to y: V = 2π ∫ y*f(y) dy. Or, if you can express y=g(x), use the Disk method with respect to x.

What happens if c or d are negative and n=-0.5?

If n=-0.5, the integral involves ln|y|. If c or d are negative or zero, ln|y| becomes undefined at y=0 or involves complex numbers if y<0. The calculator assumes c,d > 0 for n=-0.5.

Can this calculator handle improper integrals?

No, this calculator requires finite limits c and d.

What units is the volume in?

The volume will be in cubic units corresponding to the units used for y (and implied for x through k). If y is in cm, volume is in cm³.

Why use the Disk method here?

When rotating a region bounded by x=f(y) and the y-axis (x=0) around the y-axis, the cross-sections perpendicular to the y-axis are disks, making the Disk method natural.

Is the find the volume rotated about y axis calculator always accurate?

For the function x=ky^n, yes, provided the integration is performed correctly (as done here, handling n=-0.5). For more complex functions, numerical integration might be needed, introducing approximations.

Related Tools and Internal Resources

• Disk Method Calculator: A tool specifically for using the disk method for volumes of revolution.
• Shell Method Calculator: Calculate volumes using the cylindrical shells method, useful for rotation about y-axis with y=f(x).
• Definite Integral Calculator: Calculate definite integrals of various functions.
• Area Between Curves Calculator: Find the area enclosed between two functions.
• Volume of a Cylinder Calculator: Basic volume calculation.
• Derivative Calculator: Find derivatives of functions.