Question

FIZT Points) TVY NUTES DETAILS LARCALCT 14.6.055. The figure shows the region of integration for the given integral. Rewrite the integral as an equivalent iterated integral in STYLE dz dx dy x 20 y 20 z 20 dz dx dy = dz dy dx JO dy dx dz dy dx dz 42 dy dz dx VA-X dy dz dx

Answer 1

We need nest 5 eduivalent iterated 2-0 integral of 2 04-9 pa ghost dz dx dy 720 x=0 720 y 70 ZO. XZ Lel, the region of integration is D. Then descriptions of Dare the following - D-foxy s2, 05*54505252- = {04 x 54 024 & 24x, 04222-4} {05242, 05459-4, 0<x<4-y} - 205752, 04742-7. •* **4-33 plane of intersection of x24-92 x = 42 - 2² [9 = 27, & x2 42 y varies inside D, region x= 47-2² to the Cylinder x: 4-g² But in region x = 42-22, y o to the plane Thus, D= {OLELL, 04x2 42 - 2², UOL ZL2, 42-2² £ X L 64, 02 y = 14* {0:X 2 * = 4 04 752-14-x of y£ 14-x} U 02024, 2-14-* £ 242, 2-7} The projection onto the ng and 2 = 2-y, Comesponds to in the above from o the below the Curve Varies from x=ty O and, D = ocy z 2-z?

where, ZL 2-14-x comes from saling for z in x 2 4z-z? on 2 2 - 4Z +4 = 4 x . (2-2) ² 2 - 1 4 -2. Z - 2-14-2 OV We choose regeline rool, because it comesponds to A 2-z = y = 14-x Now the skenated integrals are 2 4-92 29 de dx dy y = 0 x=0 zzo 64 2-4 $ de dy da. x=0 Y=o Zzo 4-y² e2-Z 2 S dx dy dzi z=0 Yzo M- O 2 22-Y 4-4 SS dx de dy در) yao Z=o x=0 4 Z-z2 4 = SP s dydz dz + pore dy dzdz zzo O Y=0 Z=0 2. x=42-2² y=0 2-14-2 4-0 2. 2-7 S'S 04 f dy dz dzdx dydz dx. x=0 Z=0 y = 0 x=0 Z: 2-24-2 Yao