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C4 Integration Problem - Volume 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. (this question has been asked before but I still don't understand how to do it :/)

Look at the diagram below. The red curve is C, the blue line is the tangent, and the purple area is the shaded area you want to find (or rather, just obtain the limits from - we are interested in the volume, not the area!). Keep in mind that to get the shaded purple area you want to integrate the tangent line from the point P to B, and then subtract the integral of C from P to A.

So, work out what A and B are first in terms of

a

, then you got your limits for the solid of revolution.

(edited 6 years ago)

C4 Integration Problem - Volume of Revolution

RDKGames

Study Forum Helper

Original post by miadiz

a

, then you got your limits for the solid of revolution.

(edited 6 years ago)

Reply 1

brainmaster

Original post by miadiz

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.(this question has been asked before but I still don't understand how to do it :/)

use the x coordinate of the tangent and the x intercept of the curve c as limits...
let me know if you are still stuck :smile:

Edit: forgot to say that use the equation pi*integral (y^2 - y1^2) where y and y1^2 are the equations of the two lines the shaded region is between.

(edited 6 years ago)

Reply 2

miadiz

Original post by RDKGames

a

, then you got your limits for the solid of revolution.

Thank you. I get A as (a,0) and B as (5/4a, 0).
So how do I integrate the tangent line from P to A and from P to B? Sorry I just don't get how that would be written out as.

Reply 3

RDKGames

Study Forum Helper

Original post by miadiz

Thank you. I get A as (a,0) and B as (5/4a, 0).
So how do I integrate the tangent line from P to A and from P to B? Sorry I just don't get how that would be written out as.

Think about it.

You are finding the volume of the cone that is generated by rotating the tangent line about the x-axis between the point P and the point B.
You tackle it in two ways; either use the formula for the volume of a cone, or do the whole integration thing. I prefer the former.

Then you want to subtract the volume of the solid that is generated by rotating C between P and A. So, you got