Anish Patel

Posted on Friday, 06 June, 2003 - 09:10 am:

hi, i was just looking through one of dr. siklos' booklets and for one of the questions he seems to quote the result

'integral of 2*pi*y*x dx'

for the volume formed by rotating a curve about the y axis,i've never seen this-is it something i should know?i think we got taught a formula in P2, something like 'integral of pi*f(y) squared dy',any difference? Any clarification would be much appreciated,thanks!

Anish

Ian Short

Posted on Friday, 06 June, 2003 - 10:24 am:

Was the curve y=x² ?

Alice Thompson

Posted on Friday, 06 June, 2003 - 10:40 am:

I think we're looking at question 22:

The function $f$ satisfies the condition $f' (x) > 0$ for $a \leq x \leq b$ , and $g$ is the inverse of $f$ . By making a suitable change of variable, prove that

$\int_{a}^{b} f (x) dx = b β - a α - \int_{a}^{b} g (y) dy$

Where $α = f (a)$ and $β = f (b)$ . Interpret this result geometrically, by means of a sketch, in the case where $α$ and $a$ are both positive.

Prove similarly and interpret the formula

$2 π \int_{a}^{b} x f (x) dx = π (b^{2} β) (a^{2} α) - π \int_{a}^{b} [g (y {)]}^{2} dy$

Ian Short

Posted on Friday, 06 June, 2003 - 02:22 pm:

You won't have to know this formula off by heart, only understand the geometric significance of it.

Have you answered all parts of the question except this? I'm guessing that you have got to the final 'interpret the formula' bit and have drawn a suitable graph. Consider the area below the graph, above the $x$ -axis and between $y = a$ and $y = b$ . Imagine rotating this about the $y$ -axis. You have an annular base and varying height. In the formula $\int_{a}^{b} 2 π x f (x) dx$ ,

- $f (x)$ is the height

- $dx$ is width of infinitesimally small annulus at $x$

- $2 π x$ is the circumference of this infinitesimally small annulus.

Multiply them for a small slice of volume and the integral sums these volumes. That's the gist of things.

Ian

Anish Patel

Posted on Saturday, 07 June, 2003 - 10:30 am:

Ah,makes sense,cheers Ian