Question

Use triple integrals to find the volume of the right pyramid: 9 2 4 Hint: choose the coordinate system conveniently, describe the region starting from z and use similar triangles.

Answer 1

(pok with point point. I at B C اسیا GE 2 Sol (0,0,9) Let us orient this pyramid in a 3-D coordinate syster origen such that I F lies in xtanis, I J lis in positive yarisy. at 1080) & I À lies in & I A lies in positive z-ares. TO? (130) Then, due to symmetry we can just find the ola Geenidensit & multiply it by 4 to get volume of So, Consides one faiesth past fourth part of this pügamidrs AIFEJ It is bounded by 2 y z plane , xz plane, xy plane & two other planes. 2 (1,0,0) volume in I totale pyramid. CS Scanned with CamScanner

& 9C- O As for plane AFE, let ant by + cz =d Then, (6,0,2), (1,0,0), (1,2,0)) passes the it... =d a=d, atab=d gcza d=a, 2b =) plane in ax. 10 »4 2=9 is forplane A EJ, let pn+qy + r 2 = 8 Them; t(0,0,9,12,01, 10,230) ) passes the sot. axt.a ga=s þ + 2q = s. P + s =s , q 29 = s r = 3 g 2 => 1 = 2 / 3 q = 2/2, p = 0 plane is y z = 8 2 y + az 18 So, bounds of plane are in XY, 2, 2x planes, In+z=9 plane gy +22=18 plane Reg. volume 4 * 2=0X=0 1 I dudy dz 4 * 6 E24 cubic units Beste CS Scanned with CamScanner

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