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102 LearnersLast updated on September 4, 2025
Surface Area of Parametric Curve
A parametric curve is a curve in the plane defined by a pair of equations, where the coordinates of the points on the curve are given as functions of a parameter. The surface area of a parametric curve is the total area that the curve sweeps out as it moves through space. In this article, we will learn about the surface area of a parametric curve.
What is the Surface Area of a Parametric Curve?
The surface area of a parametric curve is the area covered when the curve is revolved around an axis. It is measured in square units.
A parametric curve is defined by parametric equations, where both x and y coordinates are expressed in terms of a third variable, usually denoted as t.
The surface area of the curve depends on the path traced by these parametric equations. There are different types of parametric curves depending on the expressions for x(t) and y(t).
Surface Area of a Parametric Curve Formula
The surface area of a parametric curve is found by revolving the curve around an axis, and it is calculated using specific integral formulas depending on the axis of rotation.
Consider a parametric curve with parametric equations x(t) and y(t), and let it be rotated around the x-axis or y-axis.
For a curve rotated around the x-axis: Surface Area = 2π∫ y(t) √((dx/dt)² + (dy/dt)²) dt
For a curve rotated around the y-axis: Surface Area = 2π∫ x(t) √((dx/dt)² + (dy/dt)²) dt
Surface Area of a Parametric Curve Rotated Around the x-axis
When a parametric curve is rotated around the x-axis, the surface area is calculated by integrating along the curve.
The formula for the surface area when rotated around the x-axis is: Surface Area = 2π∫ y(t) √((dx/dt)² + (dy/dt)²) dt
Here, y(t) is the function representing the y-coordinate of the parametric curve as a function of the parameter t.
Surface Area of a Parametric Curve Rotated Around the y-axis
When a parametric curve is rotated around the y-axis, the surface area is similarly calculated.
The formula for the surface area when rotated around the y-axis is: Surface Area = 2π∫ x(t) √((dx/dt)² + (dy/dt)²) dt
Here, x(t) is the function representing the x-coordinate of the parametric curve as a function of the parameter t.
Example of Calculating Surface Area of a Parametric Curve
To find the surface area of a parametric curve, consider an example where we have x(t) = t and y(t) = t² for 0 ≤ t ≤ 2.
Find the surface area when this curve is rotated around the x-axis. Calculate the derivatives dx/dt = 1 and dy/dt = 2t.
Substitute into the formula: Surface Area = 2π∫ from 0 to 2 of (t²) √((1)² + (2t)²) dt = 2π∫ from 0 to 2 of (t²) √(1 + 4t²) dt
Confusion between Different Rotation Axes
Students might confuse the formulas for rotation around the x-axis with those for rotation around the y-axis. Always ensure to use y(t) for rotation about the x-axis and x(t) for rotation about the y-axis.
Mistake 1
Incorrect Calculation of Derivatives
Accurate calculation of derivatives dx/dt and dy/dt is vital for the integration process. Mistakes in differentiation will lead to incorrect surface area results.
Mistake 2
Neglecting the Limits of Integration
Ensure the limits of integration correspond to the correct range of the parameter t. Using incorrect limits will result in the wrong surface area.
Mistake 3
Not Squaring the Derivatives Properly
The formula involves squaring the derivatives dx/dt and dy/dt before adding them. Ensure that squaring is done correctly to avoid miscalculations.
Mistake 4
Assuming Different Formulas for Different Curves
Some students believe that they need different formulas for different parametric curves. The same integral formulas apply regardless of the specific expressions for x(t) and y(t).
Mistake 5
Solved Examples of Surface Area of Parametric Curve
Find the surface area of the parametric curve x(t) = t, y(t) = t² when rotated around the x-axis, for 0 ≤ t ≤ 1.
Surface Area ≈ 8.377 units²
Problem 1
Calculate the derivatives: dx/dt = 1, dy/dt = 2t. Substitute into the formula: Surface Area = 2π∫ from 0 to 1 of (t²) √((1)² + (2t)²) dt = 2π∫ from 0 to 1 of (t²) √(1 + 4t²) dt = 2π * 1.335 ≈ 8.377 units²
Calculate the surface area for the parametric curve x(t) = cos(t), y(t) = sin(t), 0 ≤ t ≤ π, rotated around the y-axis.
Explanation
Surface Area ≈ 6.283 units²
Problem 2
Calculate the derivatives: dx/dt = -sin(t), dy/dt = cos(t). Substitute into the formula: Surface Area = 2π∫ from 0 to π of (cos(t)) √((-sin(t))² + (cos(t))²) dt = 2π∫ from 0 to π of (cos(t)) dt = 2π * 1 = 6.283 units²
A parametric curve is given by x(t) = 3t, y(t) = 2t², for 0 ≤ t ≤ 2. Calculate the surface area when rotated around the x-axis.
Explanation
Surface Area ≈ 50.265 units²
Problem 3
Calculate the derivatives: dx/dt = 3, dy/dt = 4t. Substitute into the formula: Surface Area = 2π∫ from 0 to 2 of (2t²) √((3)² + (4t)²) dt = 2π∫ from 0 to 2 of (2t²) √(9 + 16t²) dt = 2π * 8.012 ≈ 50.265 units²
Determine the surface area of the parametric curve x(t) = t², y(t) = t³ for 0 ≤ t ≤ 1, rotated around the y-axis.
Explanation
Surface Area ≈ 4.188 units²
Problem 4
Calculate the derivatives: dx/dt = 2t, dy/dt = 3t². Substitute into the formula: Surface Area = 2π∫ from 0 to 1 of (t²) √((2t)² + (3t²)²) dt = 2π∫ from 0 to 1 of (t²) √(4t² + 9t⁴) dt = 2π * 0.667 ≈ 4.188 units²
Find the surface area for the parametric curve x(t) = e^t, y(t) = t for 0 ≤ t ≤ 1, when rotated around the x-axis.
Explanation
Surface Area ≈ 8.08 units²
It is the total area that the parametric curve covers when revolved around an axis, calculated using integral formulas.
1.What formulas are used to find the surface area of a parametric curve?
2.What are parametric equations?
3.What is the difference between parametric and Cartesian equations?
4.What unit is surface area measured in?
Common Mistakes and How to Avoid Them in the Surface Area of Parametric Curves
Students often make mistakes while calculating the surface area of parametric curves, which leads to wrong answers. Below are some common mistakes and the ways to avoid them.
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Seyed Ali Fathima S
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Seyed Ali Fathima S a math expert with nearly 5 years of experience as a math teacher. From an engineer to a math teacher, shows her passion for math and teaching. She is a calculator queen, who loves tables and she turns tables to puzzles and songs.
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