Question

Clearly construct a triple integral of the form dz dy dx to find the volume of the nose of a vehicle constructed from the paraboloid y=2(x +z) and the vertical plane y=6. But do not evaluate the integral.

Answer 1

Let us draw the paraboloid : y = 2(x2+22) at-axis 8=2(x+23 B JA y=6 → X-axis 010) •Rix²+z²23 Cixy2 = 3, Z-Axis. " Volume integral ein the fortem dz dxdx cis V=S55 dzdXdx D :: For Z-dimit, take a line obarallel to z-oris feom negative to positive direction, cinside Di wherever this line enters in D is lower limit for ² and duthenter this cline is upper limit for Z. %=2(x+22) ► z²= y - x² Z= 2 2 - x² leaves D :: 2 자

Z= Thus za dimits are: 17 / 2 - x² do Z= 2 2 2 - x² Now take projection of region o shown in the the xy-plane, which looks like: 제 figure above 'Y=2x² Line y=6 NA Region Ro х B(-13,09 A (53,0) For G-limit, take a line aparallel to 4-asis. So grelinets are y=2x² to y=6. For X-limits, take all possible variations of that line inside region Ro, we get x-limits: x=-13 to x=83. Hence Volume integral is : - xe 3 6 VE dZ dy dx. X=-55 Y=2x² Z=-S-