Question

1)(a) Approximate the value of the double integral, ∫ ∫ R x 2 ydA, where R = [−1, 5] × [0, 4], using the midpoint rule with m = 3 and n = 2. (b) Evaluate the double integral in the part (a), evaluating the corresponding iterated integral.

2)Let D be a region in the xy plane, between the graphs of y = 2 cos(x) and y = − sin(x), for 0 ≤ x ≤ π 2 . Sketch D and next evaluate ∫∫ D ydA

3)Sketch a region D which is in the first quadrant of the xy plane and is bounded by the lines y = 2 − 2x and y = 4 − x. Next use the double integral over D to find the area of D.

4)Let D be a part of the disc of radius 1 and centered at the origin, where x ≥ 0. Sketch D and find the volume of the solid E which is over D and under the plane given by the equation, z = 2 − x, using the double integral in polar coordinates.

Answer 1