A set of data collected to answer a statistical question has a distribution which can be described by its __________.
2 answers:
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The expression on the left side describes a parabola. Factorize it to determine where it crosses the y-axis (i.e. the line x = 0) :
-3y² + 9y - 6 = -3 (y² - 3y + 2)
… = -3 (y - 1) (y - 2) = 0
⇒ y = 1 or y = 2
Also, complete the square to determine the vertex of the parabola:
-3y² + 9y - 6 = -3 (y² - 3y) - 6
… = -3 (y² - 3y + 9/4 - 9/4) - 6
… = -3 (y² - 2•3/2 y + (3/2)²) + 27/4 - 6
… = -3 (y - 3/2)² + 3/4
⇒ vertex at (x, y) = (3/4, 3/2)
I've attached a sketch of the curve along with one of the shells that make up the solid. For some value of x in the interval 0 ≤ x ≤ 3/4, each cylindrical shell has
radius = x
height = y⁺ - y⁻
where y⁺ refers to the half of the parabola above the line y = 3/2, and y⁻ is the lower half. These halves are functions of x that we obtain from its equation by solving for y :
x = -3y² + 9y - 6
x = -3 (y - 3/2)² + 3/4
x - 3/4 = -3 (y - 3/2)²
-x/3 + 1/4 = (y - 3/2)²
± √(1/4 - x/3) = y - 3/2
y = 3/2 ± √(1/4 - x/3)
y⁺ and y⁻ are the solutions with the positive and negative square roots, respectively, so each shell has height
(3/2 + √(1/4 - x/3)) - (3/2 - √(1/4 - x/3)) = 2 √(1/4 - x/3)
Now set up the integral and compute the volume.
Substitute u = 1/4 - x/3, so x = 3/4 - 3u and dx = -3 du.
since the hypotenuse is just the radius unit, is never negative, so the - in front of 8/17 is likely the numerator's, or the adjacent's side
now, let us use the pythagorean theorem, to find the opposite side, or "b"
so... which is it then? +15 or -15? since the root gives us both, well
angle θ, we know is on the 3rd quadrant, on the 3rd quadrant, both, the adjacent(x) and the opposite(y) sides are negative, that means, -15 = b
so, now we know, a = -8, b = -15, and c = 17
let us plug those fellows in the double-angle identities then