Question

11. Use the change of variables formula for double integrals to calculate the following double integrals over the given region D of the plane: (a) ?_D x^2 dxdy D = quarter disk bounded by the positive x axis, the positive y axis and the circle x^2 + y^2 = 9 (you may assume that cos^2 ? = 1/2(1 + cos 2?); (b) ?_D x / (x^2 + y^2)^{3/2} dxdy D = portion of annulus bounded by the circles x^2 + y^2 = 1 and x^2 + y^2 = 4 in the first quadrant x ? 0, y ? 0; (c) ?_D ydxdy D = region bounded by the x axis and the portion of the ellipse x^2/4 + y^2/9 = 1 lying in the upper half plane, y ? 0. (Note: For regions involving an ellipse whose equation is x^2/a^2 + y^2/b^2 = 1, an appropriate change of variables is x = ar cos ? y = br sin ?, 0 ? r < ?, 0 ? ? ? 2?).

          11. Use the change of variables formula for double integrals to calculate the following double integrals over the given region D of the plane:

(a) ?_D x^2 dxdy D = quarter disk bounded by the positive x axis, the positive y
axis and the circle x^2 + y^2 = 9 (you may assume that
cos^2 ? = 1/2(1 + cos 2?);

(b) ?_D x / (x^2 + y^2)^{3/2} dxdy D = portion of annulus bounded by the circles x^2 + y^2 = 1
and x^2 + y^2 = 4 in the first quadrant x ? 0, y ? 0;

(c) ?_D ydxdy D = region bounded by the x axis and the portion of the
ellipse x^2/4 + y^2/9 = 1 lying in the upper half plane, y ? 0.

(Note: For regions involving an ellipse whose equation is x^2/a^2 + y^2/b^2 = 1, an appropriate
change of variables is x = ar cos ? y = br sin ?, 0 ? r < ?, 0 ? ? ? 2?).

11. Use the change of variables formula for double integrals to calculate the following double integrals over the given region D of the plane:

(a) ?D x^2 dxdy D = quarter disk bounded by the positive x axis, the positive y
axis and the circle x^2 + y^2 = 9 (you may assume that
cos^2 ? = 1/2(1 + cos 2?);

(b) ?D x / (x^2 + y^2)^3/2 dxdy D = portion of annulus bounded by the circles x^2 + y^2 = 1
and x^2 + y^2 = 4 in the first quadrant x ? 0, y ? 0;

(c) ?D ydxdy D = region bounded by the x axis and the portion of the
ellipse x^2/4 + y^2/9 = 1 lying in the upper half plane, y ? 0.

(Note: For regions involving an ellipse whose equation is x^2/a^2 + y^2/b^2 = 1, an appropriate
change of variables is x = ar cos ? y = br sin ?, 0 ? r < ?, 0 ? ? ? 2?).

Added by Andre P.

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