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Surface Area - Multiple Integrals | Calculus - Mathematics PDF Download

In this section we will look at the lone application (aside from the area and volume interpretations) of multiple integrals in this material. This is not the first time that we’ve looked at surface area We first saw surface area in Calculus II, however, in that setting we were looking at the surface area of a solid of revolution. In other words, we were looking at the surface area of a solid obtained by rotating a function about the x or y axis. In this section we want to look at a much more general setting although you will note that the formula here is very similar to the formula we saw back in Calculus II.
Here we want to find the surface area of the surface given by z = f ( x , y ) where ( x , y ) is a point from the region D in the x y -plane. In this case the surface area is given by,
Surface Area - Multiple Integrals | Calculus - Mathematics
Let’s take a look at a couple of examples.
Example 1 Find the surface area of the part of the plane 3x+2y+z=63x+2y+z=6 that lies in the first octant.
Solution: Remember that the first octant is the portion of the xyz-axis system in which all three variables are positive. Let’s first get a sketch of the part of the plane that we are interested in.
Surface Area - Multiple Integrals | Calculus - Mathematics      Surface Area - Multiple Integrals | Calculus - Mathematics
We’ll also need a sketch of the region D.
Surface Area - Multiple Integrals | Calculus - Mathematics
Remember that to get the region D we can pretend that we are standing directly over the plane and what we see is the region D . We can get the equation for the hypotenuse of the triangle by realizing that this is nothing more than the line where the plane intersects the x y -plane and we also know that z = 0 on the x y -plane. Plugging z = 0 into the equation of the plane will give us the equation for the hypotenuse.
Notice that in order to use the surface area formula we need to have the function in the form z = f ( x , y ) and so solving for z and taking the partial derivatives gives,
Surface Area - Multiple Integrals | Calculus - Mathematics
The limits defining D are, 
Surface Area - Multiple Integrals | Calculus - Mathematics
The surface area is then,
Surface Area - Multiple Integrals | Calculus - Mathematics
Example 2 Determine the surface area of the part of z=xy that lies in the cylinder given by x2+y= 1.
Solution: In this case we are looking for the surface area of the part of z = x y where ( x , y) comes from the disk of radius 1 centered at the origin since that is the region that will lie inside the given cylinder. Here are the partial derivatives,
fx = y         f= x
The integral for the surface area is,
Surface Area - Multiple Integrals | Calculus - Mathematics 
Given that D is a disk it makes sense to do this integral in polar coordinates.
Surface Area - Multiple Integrals | Calculus - Mathematics

The document Surface Area - Multiple Integrals | Calculus - Mathematics is a part of the Mathematics Course Calculus.
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FAQs on Surface Area - Multiple Integrals - Calculus - Mathematics

1. What is surface area in the context of multiple integrals?
Ans. Surface area in the context of multiple integrals refers to the total area of a three-dimensional object's surface. It is calculated by integrating the magnitude of the cross product between the partial derivatives of the parametric equations that define the surface.
2. How does one find the surface area of a curved surface using multiple integrals?
Ans. To find the surface area of a curved surface using multiple integrals, we need to set up a double integral over a region in the parameter space. This involves parameterizing the surface using two variables, typically denoted as u and v, and then integrating the magnitude of the cross product between the partial derivatives of the parametric equations over the region.
3. Can surface area be negative when calculated using multiple integrals?
Ans. No, surface area cannot be negative when calculated using multiple integrals. The magnitude of the cross product between the partial derivatives of the parametric equations is always non-negative, and therefore the integral of this magnitude over a region will always yield a positive value representing the surface area.
4. What are some common applications of calculating surface area using multiple integrals?
Ans. Calculating surface area using multiple integrals has various applications in fields such as physics, engineering, and computer graphics. It is used to determine the amount of material required to construct curved surfaces, calculate heat transfer rates, analyze fluid flow over surfaces, and create realistic 3D models in computer simulations.
5. Are there any alternative methods to find surface area besides using multiple integrals?
Ans. Yes, there are alternative methods to find surface area besides using multiple integrals. One such method is to approximate the surface with a collection of flat polygons, calculate the area of each polygon, and sum them up. This approach, known as polygonal approximation, is often used in computer graphics and numerical simulations where the surface is discretized into small triangles or rectangles.
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