Find The Volume Generated By Revolving About The Y-axis Calculator

What is a Volume Generated by Revolving About the y-axis Calculator?

A volume generated by revolving about the y-axis calculator is a tool used to find the volume of a three-dimensional solid formed when a two-dimensional region, defined by functions and lines, is rotated around the y-axis. This calculator typically uses methods from integral calculus, like the Disk Method or Washer Method (when revolving around the y-axis, often using x as a function of y), to determine the volume.

Specifically, if a region is bounded by the curve x = g(y), the y-axis (x=0), and the horizontal lines y = c and y = d, and this region is revolved about the y-axis, the volume of the resulting solid can be found using the Disk Method. The volume generated by revolving about the y-axis calculator automates this calculation.

This calculator is useful for students studying calculus, engineers, physicists, and anyone needing to calculate volumes of solids of revolution around the y-axis.

Common Misconceptions

Revolving around y-axis always uses y=f(x): When revolving around the y-axis, it's often more convenient or necessary to express x as a function of y, i.e., x=g(y), and integrate with respect to y.
The formula is always the same: The specific formula (Disk or Washer method) depends on whether the region is bounded by one curve or two curves relative to the axis of revolution. For a region bounded by x=g(y) and the y-axis, revolved around the y-axis, the Disk method is used.

Volume Generated by Revolving About the y-axis Formula and Mathematical Explanation

When a region bounded by the curve x = g(y) (where g(y) ≥ 0 for c ≤ y ≤ d), the y-axis (x=0), and the lines y = c and y = d is revolved about the y-axis, the resulting solid can be thought of as a collection of infinitesimally thin disks stacked along the y-axis.

Each disk, at a height y, has a radius r = g(y) and thickness dy. The area of the face of this disk is A(y) = π * r² = π * [g(y)]². The volume of this infinitesimal disk is dV = A(y) dy = π [g(y)]² dy.

To find the total volume V, we integrate these infinitesimal volumes from y = c to y = d:

V = ∫_c^d π [g(y)]² dy = π ∫_c^d [g(y)]² dy

This is the formula for the Disk Method when revolving around the y-axis. Our volume generated by revolving about the y-axis calculator uses numerical methods to approximate this integral.

Variables Table

Variable	Meaning	Unit	Typical Range
x = g(y)	The function defining the curve, with x as a function of y.	Varies	Any valid function of y
c	The lower limit of integration along the y-axis.	Units of y	Any real number
d	The upper limit of integration along the y-axis.	Units of y	d ≥ c
V	The volume of the solid of revolution.	Cubic units	V ≥ 0
n	Number of intervals for numerical integration.	Integer	10 – 100000

Practical Examples (Real-World Use Cases)

Example 1: Volume of a Paraboloid

Suppose we want to find the volume of the solid generated by revolving the region bounded by x = √y, the y-axis, y = 0, and y = 4 about the y-axis.

g(y) = √y
c = 0
d = 4

Using the formula: V = π ∫₀⁴ (√y)² dy = π ∫₀⁴ y dy = π [y²/2]₀⁴ = π (16/2 – 0) = 8π cubic units.

Our volume generated by revolving about the y-axis calculator would approximate this value numerically.

Example 2: Volume of a Cone-like Shape

Find the volume of the solid generated by revolving the region bounded by x = 2 – y/2, the y-axis, y = 0, and y = 4 about the y-axis.

g(y) = 2 – y/2
c = 0
d = 4

V = π ∫₀⁴ (2 – y/2)² dy = π ∫₀⁴ (4 – 2y + y²/4) dy = π [4y – y² + y³/12]₀⁴ = π [(16 – 16 + 64/12) – 0] = 64π/12 = 16π/3 cubic units.

Using the volume generated by revolving about the y-axis calculator with these inputs would give a numerical result close to 16π/3.

How to Use This Volume Generated by Revolving About the y-axis Calculator

Enter the function x = g(y): In the "Function x = g(y)" field, input the expression for x in terms of y. Use standard JavaScript math functions if needed (e.g., `Math.sqrt(y)`, `Math.pow(y,2)`, `Math.sin(y)`).
Enter the Lower Limit (c): Input the starting y-value for the region in the "Lower Limit (c)" field.
Enter the Upper Limit (d): Input the ending y-value for the region in the "Upper Limit (d)" field. Ensure d is greater than or equal to c.
Set Number of Intervals: Adjust the "Number of Intervals" for the numerical integration. Higher values give more accuracy but may be slower.
View Results: The calculator automatically updates the volume and intermediate steps as you type. The primary result is the calculated volume.
Analyze Chart and Table: The chart visualizes g(y) and [g(y)]^2, while the table shows sample values used.
Reset or Copy: Use the "Reset" button to clear inputs and "Copy Results" to copy the output.

The results from the volume generated by revolving about the y-axis calculator provide the volume of the solid formed. This can be used in various applications, from understanding calculus concepts to engineering design.

Key Factors That Affect Volume Results

The function g(y): The shape of the curve x=g(y) directly determines the radius of the disks at each y-value. Larger values of g(y) lead to larger volumes.
The limits of integration (c and d): The interval [c, d] along the y-axis defines the height of the solid. A larger interval generally results in a larger volume.
The square of g(y): The volume depends on the integral of [g(y)]², so areas where g(y) is large contribute more significantly to the volume after squaring.
Axis of Revolution: This calculator is specifically for revolving around the y-axis. Revolving around a different axis would require a different formula or setup (e.g., using the Washer Method or Shell Method, and possibly integrating with respect to x). Our Volume of Solids of Revolution page discusses other methods.
Number of Intervals (Accuracy): In our numerical approximation, a higher number of intervals generally leads to a more accurate result for the volume calculated by the volume generated by revolving about the y-axis calculator, up to the limits of machine precision.
Continuity of g(y): The function g(y) should be continuous over the interval [c,d] for the integral to be well-defined in the standard sense. Our numerical method will still produce a value but its meaning is clearer for continuous functions.

Frequently Asked Questions (FAQ)

Q: What if g(y) is negative in the interval [c, d]? A: The formula V = π ∫ [g(y)]² dy squares g(y), so the sign of g(y) doesn't affect the volume element π[g(y)]²dy. However, g(y) typically represents a radius or distance, so it's usually non-negative when setting up the problem to represent the boundary of the region from the y-axis. If the region is between x=g1(y) and x=g2(y), you'd use the Washer Method. This calculator assumes the region is between x=g(y) and x=0 (the y-axis) and g(y)>=0.

Q: How does this calculator handle the integration? A: This volume generated by revolving about the y-axis calculator uses the Trapezoidal Rule, a numerical method, to approximate the definite integral of π[g(y)]² over the interval [c, d]. It divides the interval into many small subintervals (the "Number of Intervals") and approximates the area under the curve [g(y)]² on each subinterval.

Q: Can I use this calculator for revolving around the x-axis? A: No, this calculator is specifically for revolving around the y-axis using x=g(y). For revolving around the x-axis with y=f(x), the formula is V = π ∫ [f(x)]² dx, and you would integrate with respect to x. You might need a different calculator or setup.

Q: What if my region is bounded by two curves, x=g1(y) and x=g2(y)? A: If the region is between x=g1(y) and x=g2(y) (with g1(y) ≥ g2(y) ≥ 0, for instance) and revolved around the y-axis, you would use the Washer Method: V = π ∫ ([g1(y)]² – [g2(y)]²) dy. This calculator is for the Disk Method (region bounded by one curve and the y-axis). See our Washer Method Calculator for such cases.

Q: What does "NaN" or "Error" in the result mean? A: This usually means the function g(y) you entered was not valid JavaScript or resulted in an undefined mathematical operation (like square root of a negative number if y is negative and you used Math.sqrt(y) outside its domain within the limits c,d) for some y in the interval [c,d], or the limits were invalid. Check your function syntax and the limits.

Q: Why is the number of intervals important? A: Numerical integration approximates the true integral. The more intervals used, the smaller each subinterval, and generally, the closer the sum of the areas of the trapezoids (in the Trapezoidal rule) is to the actual area under the curve, leading to a more accurate volume from the volume generated by revolving about the y-axis calculator.

Q: Can I input very complex functions for g(y)? A: Yes, as long as they are valid JavaScript expressions using 'y' as the variable and standard `Math` object functions (e.g., `Math.sin(y)`, `Math.exp(y)`, `Math.log(y)`).

Q: What are the units of the volume? A: The units of the volume will be the cubic units of the linear measure used for y and g(y). If y and x=g(y) are in centimeters, the volume is in cubic centimeters. The volume generated by revolving about the y-axis calculator provides a numerical value; you add the units. Check our Integral Calculator for more on integration.

Related Tools and Internal Resources

Disk Method Calculator: Calculate volume by revolving around x or y-axis using the disk method for y=f(x) or x=g(y).
Washer Method Calculator: Find the volume when the region between two curves is revolved around an axis.
Volume of Solids of Revolution: An overview of different methods (Disk, Washer, Shell) for finding volumes.
Definite Integral Calculator: Calculate definite integrals of functions.
Calculus Calculators: A collection of calculators related to calculus concepts.
Area Under Curve Calculator: Find the area between a curve and the x-axis.

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