Pregunta
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Step 1
We are asked to find the surface area of the curve defined by rotated about the -axis over the interv . Recall the following formula for the surface area of a function of rotated about the -axis. Note t as the curve rotates in a circular manner about the -axis, the expression is the circumference of radius and the radical measures the arc length that is the width of a band,
We begin by substituting and its derivative in the surface area formula and simplifying,
Step 2
We have found the following integral for the surface area.
To evaluate the integral we will first make the substitution . We also will need the following to complete the substitution.
Step 3
We can now make the substitution and evaluate the definite integral with respect to .

Ask by Barrett Norton. in the United States
Mar 15,2025

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The surface area of the curve rotated about the -axis from to is

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The Deep Dive

As we explore the fascinating world of calculus and surface area, it’s intriguing to note that the method of calculating surface areas of revolution has roots that go back to the ancient Greeks, specifically Archimedes. He utilized mechanical means to find volumes and areas, a precursor to modern integral calculus. His work laid the foundation for future mathematicians and is a testament to how early thinkers grappled with, and advanced, our understanding of geometry.
In today’s world, the concept of surface areas revolved upon by curves finds robust applications in various fields. For example, in manufacturing, understanding surface areas is crucial for tasks such as coating materials or designing containers. Engineers use these computations to optimize designs, ensuring structures are not only aesthetically pleasing but also efficient and functional. It’s incredible how these mathematical concepts have tangible effects in our daily lives!

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