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.
Question 3 Evaluate Sſezx=3> 22x+3y dĀ where R is the region bounded by x = 0, y = 0 and x + y =1. (10 marks) R
Problem 5 [10 points] Set up integrals for both orders of integration. Use the more convenient order to evaluate the integral over the plane region R: A R region bounded by y 0, y x, x 4 R 1+x2 a) [2 points] First order b) [2 points] Second order c) [6 points] Evaluate the integral using the more convenient order Problem 5 [10 points] Set up integrals for both orders of integration. Use the more convenient order to evaluate the...
(10 points) Let R be the region in the first quadrant bounded by the x and y axes and the line y = 1 – 1. Notice R is a triangle with area 1/2 (you do not need to verify this). Find the coordinate of the centroid of R. For extra credit, determine the y coordinate without calculating an integral. (Note: If we regard R as a plate, then the centroid of R can also be thought of as the...
3. Let region R be bounded by y = 2x - x? and y = 0 on (0,2). Setup the definite integral(s) that represents the volume of the solid generated by rotating region about the y-axis. Draw a sketch to interpret your results.
show works please Q6 10 Points The region R is bounded by y = x3, x = 3 and y = -8. Find the area of the region R. Show all of your work and include a sketch of the region.
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
show works please Q9 10 Points Let R be the region bounded by the curve y = x2 + 1 and the lines x = 0, x = 1, and y = 1. (a) Set up, but do not evaluate, the volume of the solid obtained by rotating R about the x-axis. Show your work. (b) Set up, but do not evaluate, the volume of the solid obtained by rotating R about the line 2 1. Show your work. =
Problem 7 (12 points) Let R be the region in the first quadrant bounded from below by the line y = x and from above by the circle (x - 1)2 + y2 =1. Let C be the boundary of R traced counterclockwise. Use Green's theorem to find the outward flux of the field F=(yer" +2x) + (y+e *cosx j +
12 3. (10 points) A region R is bounded by the lines x plot of R is shown below. unded by the lines x = 1, r = 2, y = 0, and y = x2. A (a) (5 points) Set up a definite integral to calculate the volume of the solid formed by revolving R around the x-axis. (b) (5 points) Evaluate your integral.