C4 Integration Problem - Solid of Revolution

Hi, I'm having a lot of difficulty with the following problem:

The diagram shows the curve C with parametric equations x=asin^2 (t), y=acost, 0 ≤ t ≤ 1/2pi where a is a positive constant. The point P lies on C and has coordinates (3/4a, 1/2a)

(a) Find dy/dx, giving your answer in terms of t.

(b) Find an equation of the tangent to C at P.

(c) Show that a cartesian equation of C is y^2=a^2−ax.

The shaded region is bounded by C, the tangent at P and the x-axis. This shaded region is rotated through 2π radians about the x-axis to form a solid of revolution.

(d) Use calculus to calculate the volume of the solid revolution formed, giving your answer in the form kπa3, where k is an exact fraction

Ive been able to do everything up until part d. I know the formula is volume = pi * integral of y^2 dx. But I can't figure out what I need to change in terms of limits. Any help would be much appreciated.

Reply 1

CTArsenal

8

You don't have to change anything, they've given you the limits;

0 ≤ t ≤ pi/2

Reply 2

alexmufc1995

8

Original post by DAstonClarke

Hi, I'm having a lot of difficulty with the following problem:

The diagram shows the curve C with parametric equations x=asin^2 (t), y=acost, 0 ≤ t ≤ 1/2pi where a is a positive constant. The point P lies on C and has coordinates (3/4a, 1/2a)

Original post by CTArsenal

You don't have to change anything, they've given you the limits;

0 ≤ t ≤ pi/2

Surely you do need to change the limits? The domain is in t, whereas the volume of revolution involves integrating with respect to x (dx).

Therefore limits:
asin^2 (pi/2)
and
asin^2 (0)
(for the volume of revolution for C anyway, you'll need to adapt this for the x coordinate of the tangent)

(edited 10 years ago)

Reply 3

CTArsenal

8

Original post by alexmufc1995

Surely you do need to change the limits? The domain is in t, whereas the volume of revolution involves integrating with respect to x (dx).

Therefore limits:
asin^2 (pi/2)
and
asin^2 (0)
(for the volume of revolution for C anyway, you'll need to adapt this for the x coordinate of the tangent)

You can do it as pi x (the integral of (acost)^2 x dx/dt) and that will yield the answer needed, but yeah your method will probably work a lot more quicker than mine haha.

(edited 10 years ago)

C4 Integration Problem - Solid of Revolution

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