Math Is Fun Forum

Particles of masses 0.7kg and 0.3kg are attached to the ends of a light inextensible string which passes over a smooth pulley. The system is released from rest and the particles move vertically.
i) Find the tension in the string
ii) State the magnitude of the resultant force exerted on the pulley by the string.

Thanks in advance!!!

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!!