9. Calculate the triple integral of the function f (x, y, z) = e-(æ²+y²)(1–æ²–y²) over the region B trapped between the paraboloids z = 1 – x² – y? and 3x2 + 3y? – 1. = Z

Calculate the line integral f(3xy³ − 4x + 4y + 8) dx + (−3xy+5) dy, where C' is the rectangle with vertices (2, −4), (2, -3), (−3,−3), and (−3,-4) oriented clockwise. Enter an exact answer. Provide your answer below: f(3x³ 4x + 4y + 8) dx + (−3xy + 5) dy =

Calculate the line integral f (−x²y − 3x + 6y + 7) dx + (7xy + 1) dy, where C' is the rectangle with vertices (−2, −3), (−2, 2), (-4, 2), and (-4,-3) oriented counterclockwise. Enter an exact answer. Provide your answer below: fc(-x²y-3x+6y + 7) dx + (7xy+1) dy =

1. (+y + ) (1+? + +) 9. Integrate the function f(r,y, 2) = over the entire 3D coordinate system, i.e., all of R'.

Calculate the integral I = = • (y² + z²) dx + (z² + x²) dy + (x² + y²) dz where I is the Intersecting line of the sphere x2 + y2 + z2 = 2Rx and the cylinder x2 + y² = 2ax (0 0).

-3y2-3x²+9 SE sirsa S*** Conyert 2x - 9y dzdydx to integral of cyclındrı cal coordinate then evaluate

Find the centroid of the thin plate bounded by the graphs of g(x)=x² and f(x) = -x + 56. Use the equations shown below with 8 = 1 and M = area of the region covered by the plate. b x = -1/1S5x[f(x) - Sx[f(x) - g(x)] dx M a b y = /[²(x) - g² (x)] dx The centroid of the thin plate is (x,y), where x = (Type integers or simplified fractions.) 1 a and y =

Evaluate the line integral x +1 ds V1 + 4x² where C is the part of the parabola y = 9 – 22 from (-3,0) to (0, 9).

Show that (2x In yz – 5ye") dx –(5e* – x²y)dy +(x²+22) dz, -1 where C runs from (2,1,1) to (3,1, e), is independent of path in R° and use a potential function to evaluate the integral.

Calculate the line integral F-dr, where F = (3-2y, x-1, -2z) and C is the boundary of a triangle with vertices C (0, 0, -3), (0, 4, 5), and (2, 0, 1). Enter an exact answer. Provide your answer below: SF-dr=