Find The Volume Of The Solid Y Axis Calculator

Data Table (Sample Points)

y	g(y)	(g(y))^2
Enter values and calculate to see data.

Table shows values at the start of each interval and the end point.

Graph of g(y) and (g(y))^2

What is a Volume of Solid of Revolution (Y-axis) Calculator?

A Volume of Solid of Revolution (Y-axis) Calculator is a tool used to find the volume of a three-dimensional solid generated by revolving a two-dimensional region around the y-axis. This process is common in calculus, particularly when using the disk method or washer method for volumes of revolution where the axis of rotation is vertical.

This calculator typically takes a function `x = g(y)` that defines the boundary of the region, and the limits of integration `c` and `d` along the y-axis, and calculates the volume using integration. Our Volume of Solid of Revolution (Y-axis) Calculator uses the disk method, summing the volumes of infinitesimally thin disks stacked along the y-axis.

Who should use it? Students studying calculus, engineers, physicists, and anyone needing to calculate the volume of a shape formed by rotating a curve around the y-axis will find this Volume of Solid of Revolution (Y-axis) Calculator very helpful.

Common misconceptions include confusing rotation around the y-axis with rotation around the x-axis (which would involve a function `y = f(x)` and integration with respect to `x`), or mixing up the disk/washer method with the cylindrical shells method, especially when the axis of rotation is the y-axis but the function is given as `y = f(x)`.

Volume of Solid of Revolution (Y-axis) Calculator Formula and Mathematical Explanation

When we revolve a region bounded by `x = g(y)`, the y-axis, and the lines `y = c` and `y = d` around the y-axis, we generate a solid. If `g(y) ≥ 0` over `[c, d]`, we can use the disk method.

The solid can be thought of as a stack of infinitesimally thin circular disks, each with a radius `r = g(y)` and thickness `dy`. The area of each disk is `A(y) = π * r^2 = π * (g(y))^2`.

To find the total volume `V`, we integrate the area of these disks from `y = c` to `y = d`:

V = ∫[c, d] A(y) dy = ∫[c, d] π * (g(y))^2 dy = π * ∫[c, d] (g(y))^2 dy

Our Volume of Solid of Revolution (Y-axis) Calculator approximates this definite integral using the Trapezoidal Rule:

V ≈ π * (Δy / 2) * [ (g(c))^2 + 2*(g(c+Δy))^2 + ... + 2*(g(d-Δy))^2 + (g(d))^2 ]

where `Δy = (d – c) / n`, and `n` is the number of intervals.

Variables Table:

Variable	Meaning	Unit	Typical Range
`x = g(y)`	The function defining the radius of the disk at a given y	Function of y	Mathematical expressions involving y
`c`	Lower limit of integration along the y-axis	Units of y	Real numbers
`d`	Upper limit of integration along the y-axis	Units of y	Real numbers, d > c
`n`	Number of intervals for numerical integration	Integer	1 to 10000+
`V`	Volume of the solid	Cubic units	Non-negative real numbers

Practical Examples (Real-World Use Cases)

Let's see how our Volume of Solid of Revolution (Y-axis) Calculator works with examples.

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^2`, `y = 0`, and `y = 2` around the y-axis.

• Function `g(y) = y^2` (Input: `y*y` or `Math.pow(y,2)`)
• Lower limit `c = 0`
• Upper limit `d = 2`
• Number of intervals `n = 100` (for good accuracy)

Using the formula `V = π * ∫[0, 2] (y^2)^2 dy = π * ∫[0, 2] y^4 dy = π * [y^5 / 5] from 0 to 2 = π * (32/5) = 6.4π ≈ 20.106`.

Our Volume of Solid of Revolution (Y-axis) Calculator with these inputs would give a result very close to 20.106 cubic units.

Example 2: Volume of a Cone

Consider the region bounded by `x = (r/h)*y`, `y = 0`, and `y = h` (a line from (0,0) to (r,h) in the y-x plane). Revolving this around the y-axis generates a cone with base radius `r` and height `h`.

Let's say `r = 3` and `h = 5`. So, `g(y) = (3/5)*y = 0.6*y`.

• Function `g(y) = 0.6*y`
• Lower limit `c = 0`
• Upper limit `d = 5`
• Number of intervals `n = 100`

The formula for a cone is `V = (1/3)πr^2h = (1/3)π(3^2)(5) = 15π ≈ 47.124`.

Using the integral: `V = π * ∫[0, 5] (0.6y)^2 dy = π * ∫[0, 5] 0.36y^2 dy = π * [0.36y^3 / 3] from 0 to 5 = π * 0.12 * 125 = 15π ≈ 47.124`.

The Volume of Solid of Revolution (Y-axis) Calculator will confirm this result.

How to Use This Volume of Solid of Revolution (Y-axis) Calculator

1. Enter the Function `x = g(y)`: Input the function that defines the right boundary of the region being revolved. Use 'y' as the variable and standard JavaScript math functions like `Math.sqrt(y)`, `Math.pow(y, 2)`, `Math.sin(y)`, `Math.exp(y)`, etc., along with basic operators `+`, `-`, `*`, `/`. For example, `Math.sqrt(4-y*y)` or `2*y+1`.
2. Enter the Lower Limit (c): Input the starting y-value for the solid.
3. Enter the Upper Limit (d): Input the ending y-value for the solid. Ensure d > c.
4. Enter the Number of Intervals (n): Choose the number of intervals for the numerical integration. A higher number (e.g., 100-1000) gives more accuracy but takes slightly longer to compute.
5. Calculate: The calculator automatically updates the volume and other details as you type. You can also click the "Calculate Volume" button.
6. Read Results: The primary result is the calculated volume. Intermediate values and the data table/chart provide more insight into the calculation.
7. Reset: Use the "Reset" button to return to default values.

The Volume of Solid of Revolution (Y-axis) Calculator provides an approximation of the integral. The accuracy depends on the number of intervals 'n'.

Key Factors That Affect Volume of Solid of Revolution (Y-axis) Calculator Results

1. The Function `g(y)`: The shape of the function directly determines the radius of the disks at each y-value, thus significantly impacting the volume. Functions with larger values over the interval [c, d] will generate larger volumes.
2. The Limits of Integration (c and d): The interval [c, d] defines the height along the y-axis over which the solid is generated. A wider interval generally leads to a larger volume, assuming `g(y)` is non-zero.
3. The Number of Intervals (n): This affects the accuracy of the numerical integration (Trapezoidal Rule). More intervals generally lead to a more accurate volume approximation, especially for rapidly changing functions `g(y)`.
4. Whether `g(y)` is Always Non-Negative: The disk method formula `π * (g(y))^2` inherently squares `g(y)`, so the sign of `g(y)` doesn't affect `(g(y))^2`. However, the concept assumes `g(y)` represents a radius, usually taken as positive. If revolving a region between two curves, the washer method is needed.
5. The Axis of Revolution: This calculator is specifically for rotation around the y-axis. Rotation around the x-axis or other lines would require different formulas and setups (see our disk method calculator for x-axis).
6. Complexity of `g(y)`: Very complex or oscillatory functions `g(y)` might require a much larger `n` for accurate results from the Volume of Solid of Revolution (Y-axis) Calculator.

Frequently Asked Questions (FAQ)

Q1: What is the disk method for y-axis rotation?

A1: The disk method for y-axis rotation calculates the volume of a solid of revolution by summing the volumes of infinitesimally thin circular disks stacked along the y-axis. The radius of each disk is given by `x = g(y)`, and its volume is `dV = π * (g(y))^2 dy`. The total volume is `V = π ∫[c, d] (g(y))^2 dy`.

Q2: How is this different from rotation around the x-axis?

A2: For rotation around the x-axis, the function is usually given as `y = f(x)`, the limits are along the x-axis (a to b), and the volume is `V = π ∫[a, b] (f(x))^2 dx`. The disks are stacked along the x-axis. Our Volume of Solid of Revolution (Y-axis) Calculator is for y-axis rotation.

Q3: What if my function is given as y = f(x) and I rotate around the y-axis?

A3: If you have `y = f(x)` and rotate around the y-axis, you might need to solve for `x` in terms of `y` to get `x = g(y)` for the disk/washer method, or use the cylindrical shells method with integration with respect to x.

Q4: What is the washer method for y-axis rotation?

A4: The washer method is used when the region being revolved is between two curves, `x = g_outer(y)` and `x = g_inner(y)`. The volume is `V = π ∫[c, d] [(g_outer(y))^2 – (g_inner(y))^2] dy`. This calculator handles the disk method (inner radius = 0). See our washer method calculator for two functions.

Q5: Why does the calculator use the Trapezoidal Rule?

A5: The Trapezoidal Rule is a numerical method to approximate definite integrals when an exact analytical solution is difficult or impossible to find, or when we are working with a calculator that needs a numerical approach. The Volume of Solid of Revolution (Y-axis) Calculator uses it for general function inputs.

Q6: How accurate is the result from the Volume of Solid of Revolution (Y-axis) Calculator?

A6: The accuracy depends on the number of intervals 'n'. For most smooth functions, 100-1000 intervals give good accuracy. For functions with sharp changes, more intervals are needed.

Q7: Can I use this calculator for any function `g(y)`?

A7: You can use it for functions `g(y)` that are continuous over the interval [c, d] and can be evaluated by JavaScript's `Math` object and basic operators. Ensure correct syntax (e.g., `Math.pow(y,2)` instead of `y^2`).

Q8: What if `g(y)` is negative in some parts of [c, d]?

A8: The formula squares `g(y)`, so `(g(y))^2` is always non-negative. The volume calculated will be for the solid generated by revolving the magnitude of `g(y)`. If the region is defined by `x=0` and `x=g(y)`, and `g(y)` is negative, it means the region is to the left of the y-axis, but the volume calculation is the same due to squaring.

Related Tools and Internal Resources

• Disk Method Calculator (X-axis): Calculate volume when revolving around the x-axis using the disk method.
• Washer Method Calculator (X and Y axis): Calculate volume when the solid has a hole (region between two curves).
• Cylindrical Shells Calculator: An alternative method for finding volumes of revolution, especially useful when integrating with respect to the other variable is easier.
• Definite Integral Calculator: Calculate definite integrals of functions.
• Understanding Solids of Revolution: A guide to the concepts behind solids of revolution.
• Calculus Tutorials: Learn more about calculus concepts including integration and volumes.