Compute the following integral using a suitable method. Triple integral over V of cos[pi(x^2 +...

Question:

Compute the following integral using a suitable method.

{eq}\iiint_{V} \cos \left [ \pi(x^2 + y^2) \right ] \, \mathrm{d}x \, \mathrm{d}y \, \mathrm{d}z {/eq}, where {eq}V {/eq} is the region enclosed by {eq}x^2 + y^2 = z {/eq}, {eq}z = 1 {/eq}.

Cylindrical Coordinates:

One of the most important decisions we make when setting up a triple integral is choosing an appropriate coordinate system. Many integrals are very simple in one and very difficult in others. So we always want to be on the lookout for familiar symmetries. Here, we see those sums of two squares, which is a big indicator of axial symmetry, so we choose cylindrical coordinates. Recall

{eq}x = r \cos \theta {/eq}

{eq}y = r \sin \theta {/eq}

{eq}z = z {/eq}

{eq}r^2 = x^2+y^2 {/eq}

{eq}\theta = \tan^{-1} \frac{y}{x} {/eq}

{eq}dV = r \; dz \; dr \; d\theta {/eq}

Answer and Explanation: 1

Note that the region is bounded below by the paraboloid and above by the plane of constant {eq}z {/eq}. In cylindrical coordinates, we write it as {eq}r^2 \leq z \leq 1 {/eq}. We bound {eq}r {/eq} at the intersection of the surfaces:

{eq}\begin{align*} r^2 &= 1 \\ r &= 1 \end{align*} {/eq}

Lastly, we need to get all the way around the central axis, so {eq}\theta \in [0, 2\pi ] {/eq}. We find

{eq}\begin{align*} \iiint_{V} \cos \left [ \pi(x^2 + y^2) \right ] dV &= \int_0^{2\pi} \int_0^1 \int_{r^2}^1 \left( \cos \pi r^2 \right)\ r\ dz\ dr \ d \theta \\ &= \int_0^{2\pi} d \theta\ \int_0^1 (1 - r^2)\ r \cos \pi r^2\ dr \\ &= (2\pi)\ \int_0^1 r \cos \pi r^2 - r^3 \cos \pi r^2\ dr \\ &= 2\pi \left[ \frac1{2\pi}\sin \pi r^2 - \left( \frac{\pi r^2 \sin \pi r^2 + \cos \pi r^2}{2\pi^2} \right) \right]_0^1 \\ &= \sin \pi - \sin 0 - \frac1\pi \left( \pi \sin \pi - 0 + \cos \pi - \cos 0 \right) \\ &= \boldsymbol{ \frac2\pi \approx 0.63662 } \end{align*} {/eq}


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Cylindrical & Spherical Coordinates: Definition, Equations & Examples

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Chapter 13 / Lesson 10
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In this lesson, we introduce two coordinate systems that are useful alternatives to Cartesian coordinates in three dimensions. Both cylindrical and spherical coordinates use angles to specify the locations of points, a feature they share with 2-D polar coordinates.


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