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junior college 1 | H2 Maths
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Charles
Charles

junior college 1 chevron_right H2 Maths chevron_right Singapore

Need help with this question.

Date Posted: 4 years ago
Views: 375
J
J
4 years ago
i)
y = -3x + 6
Sub into equation 9(x - 1)² + y² = 9,
9(x - 1)² + (6 - 3x)² = 9
9x² - 18x + 9 + 36 - 36x + 9x² = 9
18x² - 54x + 36 = 0
x² - 3x + 2 = 0
(x - 1)(x -2) = 0
x = 1 or x = 2
y = -3(1) + 6 = 3 or y = -3(2) + 6 = 0
The coordinates of the points of intersection are (1,3) and (2,0)

(Or by observation, the intersection points are at the y-maximum point of the ellipse and the x-intercept of the line, which are (1,3) and (2,0) respectively)


ii)

9(x - 1)² + y² = 9
9(x - 1)² = 9 - y²
(x - 1)² = 1 - y²/9
(x - 1) = ± √(1 - y²/9)
x = 1 ± √(1 - y²/9)

Since we want to integrate the portion that is leftwards of x = 1,

x = 1 - √(1 - y²/9)

Now divide R into two sections. One is a right angled triangle from y = 3 to y = 6.

The other is the region bounded by the y axis and x = 1 - √(1 - y²/9) , from x = 0 to x = 3

The volume of revolution obtained from the triangle is actually the volume of a cone with radius 1 and height 3)

So volume of revolution

= ⅓πr²h + π ∫ x² dy (from x = 0 to x = 3)

= ⅓π(1²)(3) + π ∫ (1 - √(1 - y²/9) )² dy (y = 0 to y = 3)

= π + π(5 - 3π/2) using GC

(no need to manually integrate here since question is only asking for numerical value)

= 6π - 3π²/2

≈ 4.05 (3 s.f)

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Jason
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Hope it helps!