Find the average value of the function f(x,y)= xy over the rectangle 0 <= x <= 2 and 0 <= y <= 3
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Since the area of the rectangle is 2 * 3 = 6, the average value equals
(1/6) * ∫(x = 0 to 2) ∫(y = 0 to 3) xy dy dx
= (1/6) * ∫(x = 0 to 2) x dx * ∫(y = 0 to 3) y dy, since the bounds are constant
= (1/6) * [(1/2)x^2 {for x = 0 to 2}] * [(1/2)y^2 {for y = 0 to 3}]
= (1/6) * 2 * (9/2)
= 3/2.
I hope this helps!
(1/6) * ∫(x = 0 to 2) ∫(y = 0 to 3) xy dy dx
= (1/6) * ∫(x = 0 to 2) x dx * ∫(y = 0 to 3) y dy, since the bounds are constant
= (1/6) * [(1/2)x^2 {for x = 0 to 2}] * [(1/2)y^2 {for y = 0 to 3}]
= (1/6) * 2 * (9/2)
= 3/2.
I hope this helps!