For dxdydz
Limit will exist like this
x=0 to x=y^2,
y=0 to y=1-z,
z=0 to z=1.
For dzdydx,
limit will be like
z=0 to z=1-y,
y=sqrt(x) to y=1,
x=0 to x=1.
And finally for dydzdx,
y=sqrt(x) to y=1-z,
z=0 to z=1,
x=0 to x=1.
6. Express the triple SSSE f(, y, z) dv erated integral in three different ways dzdxdy, dxdydz and dydzdx, where E is the solid bounded by the given surfaces (Don't evaluate the integral) x = 2, y = 2, z = 0, x + y – 2z = 2
2. (13 points) Let E be the solid region bounded by the planes x = 0, y = 0, 2=0, and x+y+z=1. (a) Sketch E. (b) Set up the integral SSSe ex+y+z dV as a triple iterated integral. (c) Compute the integral.
Could you complete the other four orders of integration not listed too? Thanks! 1 point) Writef(x, y, z)dV as an iterated integral in each of the six orders of integration, where E is the region bounded by the surfaces y 4-x2 - 4z2 and y 0 b 82x) f(x, y, z)dzdydx hix.y gi(x)- 82(x) h2(x, y) , b , g2(y) h2(x,y) f(x, y, z)dzdxdy b- a- 82() gi(y)- h(x,y h2(xy) 1 point) Writef(x, y, z)dV as an iterated integral in...
8. Let E be the solid in the first octant bounded by: the plane 2x + y + z = 8, the vertical cylinder y = x2, and the coordinate planes x = 0 and z = 0. For each of the three parts below you must illustrate your solution with diagrams in 2 and 3 dimensions. Marks will be given for the quality of the diagrams and how they are able to help the reader understand the way in...
Problem 1 part II and Problem 2 part I and II Problem 1: (Short Answer) 6 pts] The region R is bounded by y 0, 0, 2, and y- 2 4 1, 3 pts] If R is revolved about the line x = 5. If an integral or sunn of integrals with respect to z is used to calculate the volume, explain whether the washer or shell method should be used II. 3 pts) Suppose that R is the base...
1. Let E be the solid region bounded above by the sphere 4 = x2 + y2 + z2 and below by the plane y-z =-2. a. Generate a 3D picture of the region E using 3D graphing software. b. Write the integral J [ f(x,y,z)dVas an iterated integral (in rectangular coordinates) in two different ways - one with integration with respect to z first, and one with integration with respect to y first.
QUESTION 4 YE is the solid region bounded above by the plane X + 3y + z =9, below by the plane Z=1, and on the sides by the x-0 and y=0 planes, ther f(x,y,z)dzdydx f(x,y,z)dzdydx Sfax,y,z)dV= c. $15/*../drdyx 0056" ****r«.V.Ddravox oo So De */ * 10..2)dravox coso *.2)draydx
13. Let E be the region bounded by the half sphere Sı: x2 + y2 + z2 = 4 (y>0) on one side, and the XZ-plane on the other (identified as S2) a) Show how you can parameterize S, and S2 so that both surfaces are oriented outwards. Draw the tangent vectors on S. b) Let the vector field F=<-y, x, z> represent a fluid flow field through the region E. Use Stoke’s Theorem to evaluate la curl F.ds. You...
Problem 3 6 pts] The region R bounded by y V, y 0, and 4 is revolved about the line y3 Calculate the volume of the solid using the washer method and simplify your final answer. -3 Problem 4: 8 pts] The region R is boud by y2 and y 8- 2 I. Set up, but do not evaluate, an integral or sum of integrals that would give the volume of the solid of revolution formed when R is revolved...
) Let E be the solid in R ^3 bounded by the unit sphere and the xz-plane, with y ≥0 Evaluate Z Z Z E y dV triple integral