Find the average value of f over region D.

f(x, y) = 7xsin(y), D is enclosed by the curves y = 0, y = x2, and x = 7

Note that the area of D is given by ∫∫ 1 dA
= ∫(x = 0 to 7) ∫(y = 0 to x^2) 1 dy dx
= ∫(x = 0 to 7) x^2 dx
= 343/3.

So, the average value of f over D is given by
(1/(343/3)) * ∫(x = 0 to 7) ∫(y = 0 to x^2) 7x sin y dy dx
= (3/343) * ∫(x = 0 to 7) -7x cos y {for y = 0 to x^2} dx
= (3/343) * ∫(x = 0 to 7) -7x cos(x^2) dx
= (3/343) * (-7/2) sin(x^2) {for x = 0 to 7}
= (-3/98) sin 49.

I hope this helps!