# Surface area of a parametric surface

• electroguy02
In summary, to find the surface area under the restriction s^2 + t^2 <= 1 for the surface with parametric equations r(s,t) = <st, s+t, s-t>, you can use the formula S = ∫∫√(r_s^2 + r_t^2 + 1) ds dt and the substitution s = cos(theta) and t = sin(theta). The surface area is equal to -4s + C, with limits of integration for s and t being -1 and 1.
electroguy02
1. Homework Statement [/b]

For the surface with parametric equations r(s,t) = <st, s+t, s-t>, find the equation of the tangent plane at (2,3,1).

Find the surface area under the restriction s^2 + t^2 <= 1.

2. Homework Equations [/b]

3. The Attempt at a Solution [/b]

I already found that the equation of the plane is -2x+3y-z=4. I just don't know what to do with that parametrization to find the surface area. I tried plugging in the values of r(s,t) into x, y, and z, then integrating after converting s and t (I tried s = cos(theta) and t = sin(theta)) to polar, but that didn't work.

Any ideas on how to get started?

Hi there,

To find the surface area under the restriction s^2 + t^2 <= 1, you can use the formula for surface area in terms of parametric equations:

S = ∫∫√(r_s^2 + r_t^2 + 1) ds dt

where r_s and r_t are the partial derivatives of r with respect to s and t, respectively.

In this case, we have r_s = <t, 1, 1> and r_t = <s, 1, -1>. So the integral becomes:

S = ∫∫√(s^2 + t^2 + 3) ds dt

To integrate this, you can use the substitution s = cos(theta) and t = sin(theta), which will give you:

S = ∫∫√(cos^2(theta) + sin^2(theta) + 3) cos(theta) dtheta dtheta

= ∫∫√(4) cos(theta) dtheta dtheta

= 4∫∫cos(theta) dtheta dtheta

= 4∫sin(theta) dtheta dtheta

= 4∫-cos(theta) dtheta

= -4cos(theta) + C

= -4cos(arccos(s)) + C

= -4s + C

Now, to evaluate this integral, we need to find the limits of integration. Since we are restricting s^2 + t^2 <= 1, this means that s and t can take on values between -1 and 1. So our limits of integration for s and t are both -1 and 1.

Therefore, the surface area under the restriction s^2 + t^2 <= 1 is:

S = -4s + C = -4(1) + 4(-1) = -8 units^2

I hope this helps! Let me know if you have any further questions. Good luck!

## 1. What is the definition of surface area for a parametric surface?

The surface area of a parametric surface is the measure of the total area covered by the surface. It is a mathematical concept used to calculate the amount of space that the surface occupies.

## 2. How is the surface area of a parametric surface calculated?

The surface area of a parametric surface is calculated by integrating the square root of the sum of the squares of the partial derivatives of the parametric equations with respect to the two independent variables.

## 3. Can the surface area of a parametric surface be negative?

No, the surface area of a parametric surface cannot be negative as it represents a physical quantity and cannot have a negative value.

## 4. What is the difference between parametric surfaces and regular surfaces when calculating surface area?

Parametric surfaces are defined by a set of parametric equations, while regular surfaces are defined by an explicit function. The calculation of surface area for parametric surfaces involves integrating the partial derivatives, while for regular surfaces, it is calculated using the derivative of the explicit function.

## 5. Are there any practical applications of calculating the surface area of a parametric surface?

Yes, calculating the surface area of a parametric surface has many practical applications in fields such as engineering, physics, and computer graphics. It is used to determine the amount of material needed for construction, analyze fluid flow over surfaces, and create realistic 3D models in computer graphics.

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