Solution.
In cylindrical coordinates, we can express the cylinder as
r=2cosθ,0≤θ≤π
and the sphere as
r2+z2=4. Then, z=±4−r2.
So,
V=0∫π0∫2cosθ−4−r2∫4−r2rdzdrdθ==0∫π0∫2cosθ20∫4−r2rdzdrdθ==0∫π0∫2cosθ2r4−r2drdθ==0∫π(−32(4−r2)23)∣02cosθdθ==320∫π(8−8sin3θ)dθ==3160∫π(1−sin3θ)dθ==316⋅410∫π(4−3sinθ+sin3θ)dθ==34(4θ+3cosθ−31cos3θ)∣0π==34(4π−3+31−3+31)==34(4π−316)==316π−964. Answer. 316π−964.
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