Find the volume of the solid obtained when the region bounded by the curve $y = \sin x , 0 \leq x \leq \pi$ and the x-axis is rotated about the y-axis.
A) $\frac { 1 } { 2 } \pi ^ { 3 }$
B) $\pi ^ { 2 }$
C) $\pi ^ { 3 }$
D) $4 \pi ^ { 2 }$
E) $2 \pi ^ { 2 }$
F) $2 \pi ^ { 3 }$
G) $\frac { 1 } { 2 } \pi ^ { 2 }$
H) $4 \pi ^ { 3 }$

Question

Find the volume of the solid obtained when the region bounded by the curve $y = \sin x , 0 \leq x \leq \pi$ and the x-axis is rotated about the y-axis.
A) $\frac { 1 } { 2 } \pi ^ { 3 }$ 
B) $\pi ^ { 2 }$ 
C) $\pi ^ { 3 }$ 
D) $4 \pi ^ { 2 }$ 
E) $2 \pi ^ { 2 }$ 
F) $2 \pi ^ { 3 }$ 
G) $\frac { 1 } { 2 } \pi ^ { 2 }$ 
H) $4 \pi ^ { 3 }$

Quizplus · Accepted Answer

The volume of the solid is calculated using the method of cylindrical shells: $ V = 2\pi \int_{0}^{\pi} x \sin x \, dx $. Solving this integral gives $ V = 2\pi^2 $.

Find the Volume of the Solid Obtained When the Region y=sin⁡x,0≤x≤πy = \sin x , 0 \leq x \leq \piy=sinx,0≤x≤π

Find the Volume of the Solid Obtained When the Region $y = \sin x , 0 \leq x \leq \pi$