Find the average value of f(x+y)=sin(x+y) over (a) the rectangle 0 ≤ x ≤ π/3, 0 ≤ y ≤ 5π/3 (b) the rectangle 0 ≤ x ≤ 2π/3, 0 ≤ y ≤ 7π/6 (a) the average value of f is (b) the average value of f is

For each given region , the average value of over is the integral of over divided by the area of . In both cases, is a rectangle so the area is trivial.

a. The area of is . The integral of over is

So the average value of over this region is 0.

b. The area of is . The integral of is

The average vale of over this region is then .

intensity of the vector is v= √37 ≈ 6.08 units and make angle ∡α ≈ 9.46° with east direction

step-by-step explanation:

required vector is consists of the two componentsvx= 2+4=6 units   and vy= 1 unit and vx ⊥ vywe will use pythagorian theorem to find intensity of the vector vv∧2 = vx∧2 + vy∧2 => v = √vx∧2 + vy∧2 = √6∧2 + 1∧2 = √36+1 = √37 ≈ 6.08 unitsthe angle ∡α between vector and east direction we wil find with tanαtanα = 1/6 => α = arc tanα = arc 1/6 => α ≈ 9.46°

hope this