The circle x^2 +y^2 = r^2 has radius r and centre O. The circle meets the positive x-axis at B. The point A is on the interval OB. A vertical line through A meets the circle at P. Let Theta=angle OPA.
The shaded region bounded by the arc PB and intervals AB and AP is rotated about the x-axis. Show that the volume V formed is given by
V= ((π r^3)/3)(2 - 3sin Θ + sin^3 Θ)
Having a bit of trouble with this one. I've tried to find A in terms of theta (using trig) for the definite integral but have come upon nothing. I'm sure there's probably a key rule or identity or something which I'm missing, either that or I've interpreted the sketch wrong. Maybe I'm just too sleepy. Any help is appreciated thank you
![IMG-2463.jpg](/data/attachments/37/37995-714da4889ed9eac5f8fa891af978a244.jpg)
![IMG-2462.jpg](/data/attachments/37/37996-f13d72bf0ae8f0b8ff37751a474fc9db.jpg)
The shaded region bounded by the arc PB and intervals AB and AP is rotated about the x-axis. Show that the volume V formed is given by
V= ((π r^3)/3)(2 - 3sin Θ + sin^3 Θ)
Having a bit of trouble with this one. I've tried to find A in terms of theta (using trig) for the definite integral but have come upon nothing. I'm sure there's probably a key rule or identity or something which I'm missing, either that or I've interpreted the sketch wrong. Maybe I'm just too sleepy. Any help is appreciated thank you
![IMG-2463.jpg](/data/attachments/37/37995-714da4889ed9eac5f8fa891af978a244.jpg)
![IMG-2462.jpg](/data/attachments/37/37996-f13d72bf0ae8f0b8ff37751a474fc9db.jpg)