Question

Find the volume of the solid generated by revolving the shaded region about the X-axis. Y y = sin x cos x

Answer 1

`\mathbf{ Volume (V) = (\pi^2)/(16) }`

Explanation:

`y = sin x .cos x \\ \\ A(x) = \pi [ sin \ x * cos \ x]^2`

`= \pi sin^2 x* cos ^2x`

`Volume (V) = \int ^(\pi/2)_(0) \pi \ sin^2 x \ cos^2x \ . dx`

`=\pi \int ^(\pi/2)_(0) \Bigg [ (1-cos \ 2x)/(2) * (1 + cos \ 2x)/(2) \Bigg ] \ . dx`

`=(\pi)/(4) \int ^(\pi/2)_(0) \ [1 - cos^2 \ 2x] \ . dx`

`=(\pi)/(4) \int ^(\pi/2)_(0) sin ^2 2x \ . dx`

`=(\pi)/(4) \int ^(\pi/2)_(0) \ (1-cos \ 4x)/(2) \ . dx`

`=(\pi)/(8) \int ^(\pi/2)_(0) \ (1-cos \ 4x) \ . dx`

`=(\pi)/(8) \Bigg [x - (sin \ 4x)/(4) \Bigg]^(\pi/2)_(0)`

`=(\pi)/(8) \Bigg [((\pi)/(2) -0)-0 \Bigg]`

`\mathbf{ =(\pi^2)/(16) }`

Answer 2

hello your question is incomplete attached below is the complete question

answer;
`(\pi ^(2) )/(16)`

Explanation:

attached below is a detailed solution to the given question above

where :

cos 2x = 1-2sin^2x

cos2x = 2 cos^2x -1

sin^2 2x + cos^2 2x = 1

1-cos^2 2x = sin^2 2x