Find the surface area of the given solid

In summary, the conversation discusses a problem in which the limits of integration are not clearly defined. The given parametric curve lies in Quadrants I and IV and is unbounded, so additional constraints are needed for the limits of integration. The provided formula for surface area also contains errors. It is recommended to seek alternative sources for information on calculating surface area.
  • #1
chwala
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Homework Statement
See attached
Relevant Equations
Parametric equations
My question is on how did they determine the limits of integration i.e ##2## and ##3## as highlighted? Thanks

1672655066320.png
 
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  • #2
The problem statement is incomplete.
 
  • #4
I meant that the problem statement did not include the limits, i.e., poorly written.
 
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  • #5
Frabjous said:
I meant that the problem statement did not include the limits, i.e., poorly written.
I agree. The graph of the parametric curve ##x = t^2, y = t^3## lies in Quadrants I and IV, and is unbounded. There has to be additional but unstated constraints for the limits of integration that are shown.
 
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  • #6
@chwala : It seems in your square root, you're using ##\frac {dx}{dt} ##twice, rather than what I believe is correct, ##\frac {dx}{dt}, \frac{dy}{dt}##
 
  • #7
WWGD said:
@chwala : It seems in your square root, you're using ##\frac {dx}{dt} ##twice, rather than what I believe is correct, ##\frac {dx}{dt}, \frac{dy}{dt}##
@WWGD This is not my working rather notes that i came across as indicated by the given internet link;

yes, there is a mistake there... it ought to be

$$Surface area (y-axis) = 2π \int_ a^b x(t)\sqrt{(x^{'})^2+(y{'})^2}$$

where

$$x^{'}=\dfrac{dx}{dt}$$

$$y^{'}=\dfrac{dy}{dt}$$
 
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  • #8
Since there seems to be two errors on that page, the missing information about boundaries, and the formula for surface area, perhaps you should look elsewhere for information on how to calculate surface area.
 
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FAQ: Find the surface area of the given solid

How do I find the surface area of a rectangular prism?

To find the surface area of a rectangular prism, you need to calculate the area of all six faces and then sum them up. The formula is: Surface Area = 2lw + 2lh + 2wh, where l is the length, w is the width, and h is the height of the prism.

What is the formula for the surface area of a cylinder?

The surface area of a cylinder can be found using the formula: Surface Area = 2πr(h + r), where r is the radius of the base and h is the height of the cylinder. This formula accounts for the areas of the two circular bases and the rectangular side that wraps around the cylinder.

How do I determine the surface area of a sphere?

The surface area of a sphere is calculated using the formula: Surface Area = 4πr², where r is the radius of the sphere. This formula gives you the total area covering the surface of the sphere.

What is the method to find the surface area of a cone?

To find the surface area of a cone, you use the formula: Surface Area = πr(r + l), where r is the radius of the base and l is the slant height of the cone. This formula includes the area of the circular base and the lateral surface area.

How do I calculate the surface area of a triangular prism?

The surface area of a triangular prism is found by calculating the area of the two triangular bases and the three rectangular faces, then summing these areas. The general formula is: Surface Area = (Base Area x 2) + (Perimeter of Base x Height), where the Base Area is the area of one triangular base, and the Height is the distance between the two triangular bases.

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