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cathelyn13

Member

Registered: 2009-07-18

Posts: 10

Integration Application: Volume generated by area bounded by two curve

Q: Find the volume generated by area bounded by two curves rotated through 360 degrees about the y-axis:
a) y = x²
y = 2- x²

b) y = x2
y2 = 8x

For these two parts, which formulae should I use?

Volume generated

= π ∫ x² dy
or = π ∫ [f(y)]² – [g(y)]² dy ?

For part a, it will be simplier if I use the first formula where I simplify the y before using the first formula:

y = 2 - x² - x²
y = 2 - 2x²
x = √ (2 - y)/2

After using the first formula, my answer is π.

However, if I use the second formula, I got diferent answer:

y = x²
x = √y

y = 2 - x²
x = √(2 - y)

V = π ∫ [f(y)]² – [g(y)]² dy
= π ∫ [√y]² - [√(2 - y)]² dy
= .....
= π (4 - 4)
= 0

For part b, it seems that I can't use the first formula, as after simplify,
y = √(8x) - x²
it's hard to change to make x the subject in terms of y.

Using the second formula for part b, I got the answer 24/7π.

So how do we know which formula to be used for which question? Because the question is involving volume generated by area bounded by two curves, shouldn't we be using the second formula only? Why using both formulas I get different answers? Please help me check if my answer is correct.

Besides I just knew I can use the shell method but I am more familiar with the disk method? Which formula should I be using and why the answer differs if I use both the methods I mentioned?

Many thanks in advance!!

glenn101

Member

Registered: 2008-04-02

Posts: 108

Re: Integration Application: Volume generated by area bounded by two curve

Hi cathelyn13,
I haven't encountered this specific type of integral volumes/areas stuff before but I'm assuming it works similar to the way I would go about it.
I would firstly work out the areas between the two curves
so using this formula;
Area∫ [f(x)]² – [g(x)]² dx
Where f(x) is the greater area.
and then find the volume of this
Volume rotated about y=pi ∫ x² dy

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"If your going through hell, keep going."

cathelyn13

Member

Registered: 2009-07-18

Posts: 10

Re: Integration Application: Volume generated by area bounded by two curve

Oh ok. Thanks!!