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Find, to four decimal places, the area of the part of the surface $ z = (1 + x^2)/(1 + y^2) $ that lies above the square $ | x | + | y | \leqslant 1 $. Illustrate by graphing this part of the surface.

Surface Area \approx 2.69588

00:04

Frank L.

Calculus 3

Chapter 16

Vector Calculus

Section 6

Parametric Surfaces and Their Areas

Harvey Mudd College

University of Michigan - Ann Arbor

University of Nottingham

Boston College

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so we're going to be using our calculator eventually. But before we do that, we want to set up this double integral. And we know that area is equal to three double integral of the square root of one plus z x squared plus z y squared d a. So with that, we know that since Z equals one plus X squared over one plus y squared, we want to take the partial derivatives of Z with respect to X. That'll be two x over one plus y squared and then the partial derivatives e with respect to why it will be a little bit more complicated at two i times, um, two y times one plus x squared over one plus y squared, squared. Then, um, if we take these values and square um, underneath the radical add one What we're gonna end up getting is the area being equal to this double integral inside the region D of the square root um, would actually be able to simplify this to move something out of the radical. It'll be 1/1 plus y squared squared times the square root of one plus Why square to the fourth plus four x squared times one plus y squared, squared plus four y squared times one plus x squared, squared D A And again, these are things where we would never want to calculate this by hand. So instead we can write on a calculator to do the work for us instead. So the only thing we need to do now is just get our bounds of integration. So we know that the region that we have is going to be, um, negative one toe, one for the X. So this is now D y DX. It'll be negative 1 to 1 for the X based on the region were given, and it's going to be the absolute value of X minus one to one, minus the absolute value of X. Andi. When we use a calculator to approximate this, what we end up getting is 2.6959 because we want to correct to four decimal places. So this will be our final answer for that surface area

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