double integration example help understand plz?

Question

hey guys i dont get this double integration example in my book
ill use S for integral
SSysin(xy)dA=S(from c to d)S(from a to b)ysin(xy)dxdy=S(from c to d)[-cos(xy)](from b to c)dy
thats the part i dont get, when it is integrated with respect to x leaving y constant how is the integral of
ysin(xy)=-cos(xy)??
book doesnt explain help please?

where did the y in front of sin go?

Answer 1

Consider what the derivative of -cos (xy) with respect to x is:

d(-cos (xy))/dx
sin (xy) d(xy)/dx
sin (xy) y

Or using u-substitution:

∫y sin (xy) dx
u=(xy), du=y dx
∫sin u du
-cos (xy)

The reason the y disappears is precisely the same reason that the 2 disappears when integrating 2x to obtain x² -- namely, the inversion of the chain rule for differentiation.

Answer 2

I always hated that myself...dont rmeember much of it, Anyway, conceptually, double integrals are usually a way to turn contour integrals into regular integrals by sorta bending the coodrinate system around the contour until there isnt one anymore. What was in your book could easily be a typo. Ask the teacher about it or something. What u have to keep in mind is that a lot of the vector calculus stuff is actually a little bit arbitrary...you are used to dealign with tangents in 2d calculus but in 3d its easy to forget that theres actually 6 or 7 different ways to define the tangent, and the only reason they tend to pick one of them is because it makes things shorter....but its kinda arbitrary...I dont know, i always took a little comfort in remembering that personally.
oh i just saw that other guys answer. I dont get it really. integral of a constant times some function teh constant doesnt vanish, right? has it been that long? haha.

Answer 3

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Answer 4

Let I = ∫ y.sin (xy) dx
Now when integrating w.r.t.x , the y is considered a constant.
I = - y.cos(x.y) / y = - cos (x.y)