Find the volume of the region bounded above by the surface 𝒛 = 𝟐 𝐬𝐢𝐧𝒙 𝐜𝐨𝐬𝒚 and below by the rectangle 𝑹

Question

Rectangle R = {𝟎 ≤ 𝒙 ≤ 𝝅/𝟐; 𝟎 ≤ 𝒚 ≤ 𝝅/𝟒}

Find the volume of the region bounded above by the surface z = 2 sin x cos y and below by the rectangle R.

Thank you so much if you answer this!

Answer 1

The volume of the region bounded above by the surface 𝒛 = 𝟐 𝐬𝐢𝐧𝒙 𝐜𝐨𝐬𝒚

and below by the rectangle 𝑹 = {𝟎 ≤ 𝒙 ≤ 𝝅/𝟐; 𝟎 ≤ 𝒚 ≤ 𝝅/𝟒} is given by

 

V = ∫₀^𝝅/𝟐 ∫₀^𝝅/_𝟒∫₀ ^{2𝐬𝐢𝐧𝒙 𝐜𝐨𝐬𝒚} dz dy dx

   =2∫₀^𝝅/𝟐 sinx dx ∫₀^𝝅/𝟒 cosy dy

   = 2(0-1)(-√2/2 + 0)

   = √2