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junior college 2 | H3 Maths
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There’s 2 split curve, is there a way to even integrate? I nd help!
Date Posted: 2 years ago
Views: 659
Alternative working to Boy Mow Chau's :
Translate the curve rightwards (in the direction of the x-axis) by 2 units so now the axis of rotation is the y-axis (i.e x = 0)
The lower bound is now 2 and the upper bound is 3.
Volume of revolution
= 2π ∫³₂ r(x) h(x) dx
= 2π ∫³₂ x · (x - 2) / (1 + 3(x - 2)² + (x - 2)³)) dx
= ⅔π ∫³₂ (3x² - 6x) / (1 + 3(x - 2)² + (x - 2)³) dx
Note : The expression is directly integrable since d/dx (1 + 3(x - 2)² + (x - 2)³)
= 6(x - 2) + 3(x - 2)²
= 3(2 + (x - 2))(x - 2)
= 3x(x - 2)
= 3x² - 6x
_____________________________________________
= ⅔π [ln(1 + 3(x - 2)² + (x - 2)³]³₂
= ⅔π [ln (1 + 3(3 - 2)² + (3 - 2)³) - ln (1 + 3(2 - 2)² + (2 - 2)³) ]
= ⅔π [ ln 5 - ln 1]
= ⅔π ln 5
Translate the curve rightwards (in the direction of the x-axis) by 2 units so now the axis of rotation is the y-axis (i.e x = 0)
The lower bound is now 2 and the upper bound is 3.
Volume of revolution
= 2π ∫³₂ r(x) h(x) dx
= 2π ∫³₂ x · (x - 2) / (1 + 3(x - 2)² + (x - 2)³)) dx
= ⅔π ∫³₂ (3x² - 6x) / (1 + 3(x - 2)² + (x - 2)³) dx
Note : The expression is directly integrable since d/dx (1 + 3(x - 2)² + (x - 2)³)
= 6(x - 2) + 3(x - 2)²
= 3(2 + (x - 2))(x - 2)
= 3x(x - 2)
= 3x² - 6x
_____________________________________________
= ⅔π [ln(1 + 3(x - 2)² + (x - 2)³]³₂
= ⅔π [ln (1 + 3(3 - 2)² + (3 - 2)³) - ln (1 + 3(2 - 2)² + (2 - 2)³) ]
= ⅔π [ ln 5 - ln 1]
= ⅔π ln 5
See 1 Answer
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shell method better for this problem.
base on the question description, the region to be rotated is only the portion between x=0 and x=1.
radius of the shell is taken as x+2 because the region is rotated around line x=-2.
height of the shell is just distance between curve and x-axis.
at first glance, the cubic denominator cannot factorise, but closer look shows that the numerator is just a multiple of the derivative of denominator, so it is simple integration here.
verified with GC. GC gives equivalent numerical value.
base on the question description, the region to be rotated is only the portion between x=0 and x=1.
radius of the shell is taken as x+2 because the region is rotated around line x=-2.
height of the shell is just distance between curve and x-axis.
at first glance, the cubic denominator cannot factorise, but closer look shows that the numerator is just a multiple of the derivative of denominator, so it is simple integration here.
verified with GC. GC gives equivalent numerical value.
Date Posted:
2 years ago
normally, after integration, if obtain ln function, should show as ln|f(x)|, with modulus sign, when it is possible that f(x) is negative.
here, for the region from x=0 to x=1, x^3+3x^2+1 is positive.
here, for the region from x=0 to x=1, x^3+3x^2+1 is positive.
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