Question

N2 Use a triple integral to find the volume of the region in the first octant beneath the plane 40 pts 3x + y + 2z = 6

Answer 1

Na - The gives equation of the plane is 3x + y + 2z = 6 From the given equation the limits are the following os x=2, osy £ 6-3x, oz za 6-30-y . The volume of the region is v=p2p 6-3x, 6-30-7 - dzdy de Integrate with respect to z, we get 2 2_ To 5616–32-7) dy dix Integrate with respect to y, we get 1162) * - 7 6-32 de $16) (6-5) – (6.3232.7 ore so (-39)”. (5 - ) dhe = 12 (6-32)² (+) dx

= (36-367 +922) deze = 12 (4-4*+22) de Integrate with respect to x, we get = [ 48 – 42 + 2372 = 3 [4.) – 2.(4) + $ = (8-8 + $) Therefore, volume = 167 cubie units