Calculus- Early Transcendentals, 2021a

516 Multiple Integration
∫∫
Exercise 14.7.2 Evaluate xydxdy over the square with corners (0,0), (1,1), (2,0), and (1,−1) in two
ways: directly, and using x =(u + v)/2,y=(u − v)/2.
∫∫
Exercise 14.7.3 Evaluate x 2 +y 2 dxdy over the square with corners (−1,0), (0,1), (1,0), and (0,−1)
in two ways: directly, and using x =(u + v)/2,y=(u − v)/2.
∫∫
Exercise 14.7.4 Evaluate (x + y)e x−y dxdy over the triangle with corners (0,0), (−1,1), and (1,1) in
two ways: directly, and using x =(u + v)/2,y=(u − v)/2.
∫∫
Exercise 14.7.5 Evaluate y(x − y)dxdy over the parallelogram with corners (0,0), (3,3), (7,3), and
(4,0) in two ways: directly, and using x = u + v, y = u.
∫∫ √
Exercise 14.7.6 Evaluate x 2 + y 2 dxdy over the triangle with corners (0,0), (4,4), and (4,0) using
x = u, y = uv.
∫∫
Exercise 14.7.7 Evaluate ysin(xy)dxdy over the region bounded by xy = 1, xy= 4, y= 1, and y = 4
using x = u/v, y = v.
∫∫
Exercise 14.7.8 Evaluate sin(9x 2 + 4y 2 )dA, over the region in the first quadrant bounded by the
ellipse 9x 2 + 4y 2 = 1.
Exercise 14.7.9 Compute the Jacobian for the substitutions x = ρ sinφ cosθ,y= ρ sinφ sinθ,z= ρ cosφ.
∫∫∫
Exercise 14.7.10 Evaluate
E
dV where E is the solid enclosed by the ellipsoid
x 2
a 2 + y2
b 2 + z2
c 2 = 1,
using the transformation x = au, y = bv, and z = cw.