Integrating parametric equations - Help A level Maths

could you please send us a picture of figure 6? Thanks

I've done that now

Have you worked out the integrand? So y dx/dtheta dtheta? Or whats the problem?

I think dx/theta=-24sinthetha cos^2(thetha)
But then I don't know how to change the limits or rearrange it into the required form

Sure, so multipy it by y and what are the limits for theta going from left to right.

I think the limits are x=0 and then I made y=0 to get theta =0 and the subbed it into x to get x=8

Youre integrating wrt theta, not x. What are the limits for theta going from left to right

I got pi/2 and 0 but I'm not sure if they're right

Looking at the ans, what might the 4 represent and why are they integrating between 0 and pi/3 - what do those points represent?
A fair bit of questions like this is to look at the ans and work back a bit. Theres a few questions you should be asking yourself
If you can get a rough answer to those before you start writing, it should be fairly straightforward.

I've got that the integral between pi/3 and 0 = -144sin^3theta costheta

Sure, so youre limits are the wrong way round and your integrand is negative so ....
Also you have a single term with sin^3 in, so is there an identity you could use to put it in the form they want?

Maybe sin^2theta +Cos^2theta=1

Its easier to simply do it rather than replying here?

But I don't know how to do it

sin^2 = 1 - cos^2
and sub it in and the integrand becomes ....

-144sinthetha (1-cos^dtheta) costheta

Youve lost a cos somewhere (as your original dx/dtheta was ^2) in your working and no idea what ^dtheta represents and why its in the middle.