Surface Area of Rotated Curve y=1+5x2 | x=0 to x=8

In summary, the task is to find the area of the surface obtained by rotating the curve y=1+5x2 from x=0 to x=8 about the y-axis. The equation for finding surface area is As=\int2\pig(x)sqrt(1+[g'(x)]2), but the integral becomes too complex to solve easily. An alternative solution is using the formula ∫ 2πx √(1 + g'(x)2) dx.
  • #1
danielatha4
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0

Homework Statement


Find the area of the surface obtained by rotating the curve y=1+5x2 from x=0 to x=8 about the y-axis



Homework Equations


As=[tex]\int[/tex]2[tex]\pi[/tex]g(x)sqrt(1+[g'(x)]2)


The Attempt at a Solution


I changed the limits to suit the y direction, the lower limit becomes 1 and the upper 321. I solved x as a function of y in order to rotate around the y-axis and when I plug everything into the formula is becoems a really messy integral I can't solve. I'm sure there's an easier way.
 
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  • #2
Hi danielatha4! :smile:

Why not just use ∫ 2πx √(1 + g'(x)2) dx ?
 

FAQ: Surface Area of Rotated Curve y=1+5x2 | x=0 to x=8

1. What is the formula for finding the surface area of a rotated curve?

The formula for finding the surface area of a rotated curve is SA = 2π ∫ a^b y√(1+(dy/dx)^2) dx, where a and b are the bounds of the curve and dy/dx represents the derivative of the curve.

2. How do you find the bounds of the curve for calculating surface area?

The bounds of the curve can be found by setting the given equation equal to zero and solving for x. In this case, for the curve y=1+5x^2, the bounds would be x=0 and x=8.

3. What does the variable 'y' represent in the formula for surface area?

The variable 'y' represents the y-value of the curve at a specific point. It is used to determine the height of each infinitesimal rectangle that makes up the surface area.

4. Can the formula for surface area of a rotated curve be applied to any curved shape?

Yes, the formula can be applied to any curved shape as long as its equation is known and it can be rotated around a horizontal or vertical axis.

5. Is the calculated surface area of a rotated curve always an exact value?

No, the calculated surface area is an approximation as it is impossible to have an infinite number of infinitesimal rectangles to perfectly cover the curve.

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