Use the given transformation to evaluate the integral. double integral (x-3y)dA, where R is the triangular region with vertices (0, 0), (2, 1) and (1, 2); x=2u+v, y=u+2v

Use the given transformation $x = 2u + v, y = u + 2v$ to evaluate the integral $\iint_{R}(x - 3y)~dA$ where $R$ is the triangular region with vertices (0, 0), (2, 1), and (1, 2).

Find an equation of the tangent plane to the given parametric surface at the specified point.

$$x=u^2+1, y=v^3+1, z=u+v; (5, 2, 3)$$

Show that F is conservative and use this fact to evaluate ∫cF·dr along the given curve.

$$F(x, y) = (4x^3y^2-2xy^3)i+(2x^4y-3x^2y^2+4y^3)j$$

,

$$C:r(t)=(t+sinπt)i+(2t+cosπt)j, 0≤t≤1$$

Use the given transformation to evaluate the integral. double (x^2-xy+y^2)dA, where R is the region bounded by the ellipse x^2-xy+y^2=2; x=2^1/2u-(2/3)^1/2v, y=2^1/2u+(2/3)^1/2v

Evaluate the double integral over the given region R.

$$\iint _ { R } x y e ^ { xy ^ { 2 } } d A , \quad R : 0 \leq x \leq 2 , \quad 0 \leq y \leq 1$$

Use the given transformation to evaluate the integral. double integral xydA, where R is the region in the first quadrant bounded by the lines y=x and y=3x and the hyperbolas xy=1, xy=3; x=u/v, y=u

Evaluate the integral by making an appropriate change of variables. double integral cos(y-x/y+x)dA, where is the trapezoidal region with vertices (1, 0), (2, 0), (0, 2), and (0, 1)

Find the mass and center of mass of the lamina that occupies the region D and has the given density function rho. D is the triangular region with vertices (0, 0), (2, 1), (0, 3); rho(x,y)=x+y

Use the transformation u=x+y, v=x-y to find

$$\iint _ { R } ( x - y ) e ^ { x ^ { 2 } - y ^ { 2 } } d A$$

over the rectangular region R enclosed by the lines x+y=0, x+y=1, x-y=1, x-y=4.

Express as a region of type I and also as a region of type II. Then evaluate the double integral in two ways. double integral xdA, D is enclosed by the lines y=x, y=0, x=1

A lamina with constant density p(x, y) = p occupies the given region. Find the moments of inertia Ix and Iy and the radii of gyration ^=x and ^=y. The triangle with vertices (0, 0), (b, 0), and (0, h)

Use the change of variables formula and an appropriate transformation to evaluate ∫∫g xy dA, where R is the square with vertices (0, 0), (1,1), (2, 0), and (1, -1).

Evaluate the double integral over the region $R$ that is bounded by the graphs of the given equations. Choose the most convenient order of integration.

$\iint_R x e^y d A ; R$ the same as in earlier problem.