change the order of integration, make the drawing area of integration
(integral from 0 to 3 dx ) integral from (x^2) to (x+6) f(x,y)dy
(integral from 0 to 3 dx ) integral from (x^2) to (x+6) f(x,y)dy
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The given region is between y = x^2 to y = x+6 for x in [0, 3].
This may be rewritten in two parts:
(i) x = y - 6 to x = sqrt(y) for y in [6, 9]
(ii) x = 0 to x = sqrt(y) for y in [0, 6].
(sketching the region should make this clear).
So, the integral equals
∫(y = 0 to 6) ∫(x = 0 to sqrt(y)) f(x,y) dx dy + ∫(y = 6 to 9) ∫(x = y-6 to sqrt(y)) f(x,y) dx dy.
I hope this helps!
This may be rewritten in two parts:
(i) x = y - 6 to x = sqrt(y) for y in [6, 9]
(ii) x = 0 to x = sqrt(y) for y in [0, 6].
(sketching the region should make this clear).
So, the integral equals
∫(y = 0 to 6) ∫(x = 0 to sqrt(y)) f(x,y) dx dy + ∫(y = 6 to 9) ∫(x = y-6 to sqrt(y)) f(x,y) dx dy.
I hope this helps!