Answer:
[tex]\mathbf{ Volume (V) = \dfrac{\pi^2}{16} }[/tex]
Step-by-step explanation:
[tex]y = sin x .cos x \\ \\ A(x) = \pi [ sin \ x * cos \ x]^2[/tex]
[tex]= \pi sin^2 x* cos ^2x[/tex]
[tex]Volume (V) = \int ^{\pi/2}_{0} \pi \ sin^2 x \ cos^2x \ . dx[/tex]
[tex]=\pi \int ^{\pi/2}_{0} \Bigg [ \dfrac{1-cos \ 2x}{2} \times \dfrac{1 + cos \ 2x}{2} \Bigg ] \ . dx[/tex]
[tex]=\dfrac{\pi}{4} \int ^{\pi/2}_{0} \ [1 - cos^2 \ 2x] \ . dx[/tex]
[tex]=\dfrac{\pi}{4} \int ^{\pi/2}_{0} sin ^2 2x \ . dx[/tex]
[tex]=\dfrac{\pi}{4} \int ^{\pi/2}_{0} \ \dfrac{1-cos \ 4x}{2} \ . dx[/tex]
[tex]=\dfrac{\pi}{8} \int ^{\pi/2}_{0} \ (1-cos \ 4x) \ . dx[/tex]
[tex]=\dfrac{\pi}{8} \Bigg [x - \dfrac{sin \ 4x}{4} \Bigg]^{\pi/2}_{0}[/tex]
[tex]=\dfrac{\pi}{8} \Bigg [(\dfrac{\pi}{2} -0)-0 \Bigg][/tex]
[tex]\mathbf{ =\dfrac{\pi^2}{16} }[/tex]
