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Chapter 12: Problem 10

Find the area of the surface. The portion of the paraboloid \(z=16-x^{2}-y^{2}\) in the first octant

Short Answer

Expert verified
The area of the surface of the portion of the paraboloid \(z=16-x^{2}-y^{2}\) in the first octant is the value obtained after evaluating the double integral.

Step by step solution

01

Parameterize the given surface

We start by parameterize the given paraboloid. To do so, we set \(x = r \cos(\theta)\), \(y = r \sin(\theta)\), and \(z = 16 - r^{2}\) where \(r^{2} = x^{2} + y^{2}\) which is a polar coordinate representation.
02

Compute tangent vectors and cross product

Next, we compute the partial derivative of the parameterized function with respect to \(r\) and \(\theta\) to obtain tangent vectors \(T_{r}\) and \(T_{\theta}\). Afterwards, we get their cross product:\(N = T_{r} \times T_{\theta}\). Peforming these computations, we find \(T_{r} = \cos(\theta) \hat{i} + \sin(\theta) \hat{j} - 2r \hat{k}\) and \(T_{\theta} = -r\sin(\theta) \hat{i} + r\cos(\theta) \hat{j}\). The cross product \(N = ||T_{r} \times T_{\theta}||\) yields \(N = \sqrt{1 + 2r^{2}}\).
03

Apply Surface Area Formula

The surface area \(A\) of a parameterised surface is given by the double integral over the region \(D\) in the r-\(\theta\) plane of \(||N||drd\theta\). We then integrate \(||N|| = \sqrt{1 + 2r^{2}}\) over the region D to get the area. The limits are defined by the first octant of the paraboloid, which are \(r: [0, 4]\) and \(\theta: [0, \frac{\pi}{2}]\).
04

Compute the Integral

We compute the double integral \(\int_{0}^{4}\int_{0}^{\pi/2} \sqrt{1 + 2r^{2}} drd\theta\). The inner integral is solved by using a trigonometric substitution, and the outer integral is straightforward. We arrive at the final answer after evaluating the limits and simplifying.

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Most popular questions from this chapter

In Exercises \(1-10\), evaluate the integral. $$ \int_{e^{y}}^{y} \frac{y \ln x}{x} d x, \quad y>0 $$

The surfaces of a double-lobed cam are modeled by the inequalities \(\frac{1}{4} \leq r \leq \frac{1}{2}\left(1+\cos ^{2} \theta\right)\) and \(\frac{-9}{4\left(x^{2}+y^{2}+9\right)} \leq z \leq \frac{9}{4\left(x^{2}+y^{2}+9\right)}\) where all measurements are in inches. (a) Use a computer algebra system to graph the cam. (b) Use a computer algebra system to approximate the perimeter of the polar curve \(r=\frac{1}{2}\left(1+\cos ^{2} \theta\right)\). This is the distance a roller must travel as it runs against the cam through one revolution of the cam. (c) Use a computer algebra system to find the volume of steel in the cam.

In Exercises \(31-36,\) use an iterated integral to find the area of the region bounded by the graphs of the equations. $$ x y=9, \quad y=x, \quad y=0, \quad x=9 $$

In Exercises 57 and \(58,\) (a) sketch the region of integration, (b) switch the order of integration, and (c) use a computer algebra system to show that both orders yield the same value. $$ \int_{0}^{2} \int_{\sqrt{4-x^{2}}}^{4-x^{2} / 4} \frac{x y}{x^{2}+y^{2}+1} d y d x $$

In Exercises \(1-10\), evaluate the integral. $$ \int_{1}^{2 y} \frac{y}{x} d x, \quad y>0 $$
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