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AnnZ [28]
2 years ago
11

Compute the integral Z Z Z U y dV , where U is the part of the ball of radius 2, centered at (0, 0, 0), that lies in the 1st oct

ant. Recall that the first octant is the part of the 3d space where all three coordinates x, y, z are nonnegative. (Hint: You may use cylindrical or spherical coordinates for this computation, but note that the computation with cylindrical coordinates will involve a trigonometric substitution - so spherical cooridnates should be preferable.)
Mathematics
1 answer:
irga5000 [103]2 years ago
3 0

As the hint suggests, convert to spherical coordinates using

x = p cos(u) sin(v)

y = p sin(u) sin(v)

z = p cos(v)

dV = dx dy dz = p² sin(v) dp du dv

Then U is the set

U = \left\{ (p,u,v) \mid 0\le p\le2 \text{ and } 0\le u\le \dfrac{\pi}2 \text{ and } 0\le v\le\dfrac{\pi}2\right\}

and the integral of y over U is

\displaystyle \iiint_U y \, dV = \iiint_U p\sin(u)\sin(v) \cdot p^2 \sin(v) \, dV

\displaystyle \iiint_U y \, dV =  \int_0^{\frac\pi2} \int_0^{\frac\pi2} \int_0^2 p^3 \sin(u) \sin^2(v) \, dp \, du \, dv

\displaystyle \iiint_U y \, dV = \frac{2^4-0^4}4 \int_0^{\frac\pi2} \int_0^{\frac\pi2} \sin(u) \sin^2(v) \, du \, dv

\displaystyle \iiint_U y \, dV =  4 \cdot \left(-\cos\left(\frac\pi2\right) + \cos(0)\right) \int_0^{\frac\pi2} \sin^2(v) \, dv

\displaystyle \iiint_U y \, dV =  4 \cdot \frac12 \int_0^{\frac\pi2} (1-\cos(2v)) \, dv

\displaystyle \iiint_U y \, dV =  2 \left(\left(\frac\pi2 - \frac12 \sin\left(2\cdot\frac\pi2\right)\right) - \left(0 - \frac12 \sin\left(2\cdot0\right)\right) \right)

\displaystyle \iiint_U y \, dV =  \pi - \sin(\pi) = \boxed{\pi}

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Answer:

52,500

Step-by-step explanation:

We can convert the word problem into an equation:

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The previous population was 52,500!

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Laurence, the stadium manager, spends $22,000 on a broadcast TV ad, $30,000 on a cable TV ad, and $6,000 on a radio ad. What per
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38 % of amount is spent on broadcast TV advertising

<em><u>Solution:</u></em>

Given that,

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3 years ago
What are the zeros of the function f(x) = x4 − 4x2 − 5?
Gekata [30.6K]

Answer:

A graph shows zeros to be ±3. Factoring those out leaves the quadratic

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The function has roots -3, 3, 2-i, 2+i.

Step-by-step explanation:

4 0
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B. 4

Step-by-step explanation:

The given function is of the form:

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We can read from the graph that, the function ranges between:

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This implies that:

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Therefore the amplitude of the function is 4.

Hence the value of A must be 4.

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