The paraboloid meets the x-y plane when x²+y²=9. A circle of radius 3, centre origin.
<span>Use cylindrical coordinates (r,θ,z) so paraboloid becomes z = 9−r² and f = 5r²z. </span>
<span>If F is the mean of f over the region R then F ∫ (R)dV = ∫ (R)fdV </span>
<span>∫ (R)dV = ∫∫∫ [θ=0,2π, r=0,3, z=0,9−r²] rdrdθdz </span>
<span>= ∫∫ [θ=0,2π, r=0,3] r(9−r²)drdθ = ∫ [θ=0,2π] { (9/2)3² − (1/4)3⁴} dθ = 81π/2 </span>
<span>∫ (R)fdV = ∫∫∫ [θ=0,2π, r=0,3, z=0,9−r²] 5r²z.rdrdθdz </span>
<span>= 5∫∫ [θ=0,2π, r=0,3] ½r³{ (9−r²)² − 0 } drdθ </span>
<span>= (5/2)∫∫ [θ=0,2π, r=0,3] { 81r³ − 18r⁵ + r⁷} drdθ </span>
<span>= (5/2)∫ [θ=0,2π] { (81/4)3⁴− (3)3⁶+ (1/8)3⁸} dθ = 10935π/8 </span>
<span>∴ F = 10935π/8 ÷ 81π/2 = 135/4</span>
Answer:
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Step-by-step explanation:
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This should have been worth more points, but anyways, here are the answers. Please give thanks :)
1.)Y = 5 - 3X<span>
2.) Y= </span><span>3X - 4</span><span>
3.) Y = 7 - X
4.) Y = -5X
5.) Y = </span>X +6 <span>
6.) Y = 5 -3X
7.) Y = 2 +</span>

<span>X
8.) </span>Y =

X - 3<span>
9.) Y= 2 - [</span>tex] \frac{2}{3} [/tex]<span>
10.) Y = 1 - 5X
11.) Y = 2 - </span>

<span>X
12.) Y = -5 -2X
13.) Y = X - 6
14.) Y = </span>

<span> -2X
15.) Y = 2 +</span>

<span>X
16.) Y = 2 +</span>

<span>X
17.) Y = 1 - 5X
18.) Y = </span>[<span>tex] \frac{3}{4} [/tex]</span>X + 2