How do I calculate the surface area of a rotated curve?

In summary, to find the surface area of a f(x) rotated around the y axis, you can use the formula S = ∫ 2πx √(1 + (f'(x)2)) dx, where the integral is taken over the appropriate interval on the x-axis. This can be done by breaking up the interval into small pieces and approximating the curve with a line segment. The sum of these approximations can be taken in the limit to get the exact surface area. An example is provided for the function f(x) = x^2.
  • #1
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How do I find the surface area of a f(x) rotated around the y axis?
 
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  • #2
Check the site mathispower4u.com in the calc 3 section there should be a couple short tutorials on surfaces of revolution that should help.

Heres one video

 
  • #3
Imagine breaking up the interval on the x-axis (over which the curve lies that is going to be rotated) into small pieces of length Δx each.. Approximating this piece of the curve y = f(x) by a line segment, we see that if it is rotated about the y-axis we get a piece of a cone. The length of that line segment between (x, f(x)) and (x+Δx, f(x+Δx) is approximately

ΔL = √((Δx)2 + (f'(x)Δx)2))​

= √(1 + (f'(x)2)) Δx​

Do you see why? This is a good point to stop and make sure you understand the reason for the last two expressions.

Therefore the surface area of the piece of surface generated when this line segment is rotated about the y-axis is approximately

ΔS = 2πx ΔL = 2πx √(1 + (f'(x)2)) Δx.​

Adding these up for all Δx, we get

S ≈ k 2πxk √(1 + (f'(x)2)) Δx​

where xk is (say) the left endpoint of each interval of length Δx on the x-axis.

So in the limit, this becomes exactly equal to the integral

S = 2πx √(1 + (f'(x)2)) dx​

where the integral is taken over the appropriate interval on the x-axis.

As an example, try to determine the area of the surface generated by rotating the graph of the function

f(x) = x2

for 0 ≤ x ≤ 1 about the y-axis.

(If your exact answer is 5.330 when rounded to three decimal places, then you probably got the same one as I did.)
 
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What is the definition of "surface area of a manifold"?

The surface area of a manifold is a mathematical concept that measures the total area of a curved surface. In simpler terms, it is the amount of space on the surface of a three-dimensional object.

How is the surface area of a manifold calculated?

The surface area of a manifold can be calculated using various mathematical techniques, depending on the shape and complexity of the object. Generally, it involves breaking down the surface into smaller, more manageable pieces and then adding up their areas using integration.

Why is the surface area of a manifold important in science?

The surface area of a manifold is an essential concept in many scientific fields, such as physics, chemistry, and engineering. It is used to calculate the amount of heat or energy transferred during processes like heat transfer and fluid flow. It also helps in understanding the properties and behavior of objects in the real world.

What are some real-world applications of the surface area of a manifold?

The surface area of a manifold has numerous practical applications. It is used in fields like architecture, where it helps in determining the amount of materials needed to construct a building. It is also used in manufacturing processes, where it helps in designing efficient and cost-effective production methods.

What are some common misconceptions about the surface area of a manifold?

One common misconception is that the surface area of a manifold is the same as its volume. While both concepts are related, they measure different aspects of an object. Another misconception is that the surface area of a manifold only applies to 3-dimensional objects, when in fact it can also be applied to higher-dimensional objects, such as 4-dimensional spacetime in physics.

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