maths.find volume.parabola20pt

chi man

2009-10-29 6:36 am

1) Find the volume obtained when the region between the graph of y=sqrt(x^2 +1) and the x-axis, between x=3, x=4, is rotated around the x-axis.

2) Find the coordinates of the focus of the parabola y= (x^2 /8) +1.

3) Find the equation of the parabola with focus (1,3) and directrix y= -3.

回答 (1)

用戶9112

2009-10-29 6:59 am

✔ 最佳答案

(1) Volume = ∫ πy^2 dx (x from 3 to 4)
= π∫ (x^2 + 1) dx (from 3 to 4)
= π(x^3/3 + x) (from 3 to 4)
= π(64/3 + 4 - 9 - 3)
= 40π/3
(2) Let the focus be (0, f)
The vertex is (0, 1)
Then the directrix is y = 1 - (f - 1) = 2 - f
Distance of point (x,y) from directrix = distance from (1,f) =>
| y - 2 + f | = √(x^2 + (y - f)^2
y^2 + 4 + f^2 - 4y - 4f + 2fy = x^2 + y^2 - 2fy + f^2
(4f - 4)y = x^2 + 4f - 4
y = x^2/(4f - 4) + 1 => 4f - 4 = 8 => f = 3
Therefore the focus is (0,3)
(3) | y + 3 | = √[(x - 1)^2 + (y - 3)^2]
y^2 + 6y + 9 = (x - 1)^2 + y^2 - 6y + 9
12y = (x - 1)^2
y = (x - 1)^2 / 12

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