I need someone REALLY good at math, please :)?

Question

Hello, I have some multivariable problems that can be done fairly easily, but I really need help! Here is the problem:

Evaluate the integral by changing the order of integration in an appropriate way:

(It's a triple integral)

int(0 to 2) int(0 to 4-x^2) int (0 to x) (sin 2z / (4-z)) dydzdx

Thank you very much in advance!

Answer 1

The bounds of integration are 0
[0, 2]∫ [0, √(4-z)]∫ [0, x]∫ sin (2z)/(4-z) dy dx dz

Evaluating the inner integral (easy, since this is just a constant in terms of y):

[0, 2]∫ [0, √(4-z)]∫ x sin (2z)/(4-z) dx dz

Evaluating the second integral (again easy, since this is just a linear function):

[0, 2]∫1/2 x² sin (2z)/(4-z) |[0, √(4-z)] dz
[0, 2]∫1/2 √(4-z)² sin (2z)/(4-z) dz
[0, 2]∫1/2 sin (2z) dz

And with that lovely cancellation, we evaluate the last integral:

1/4 (- cos (2z)) | [0, 2]
-1/4 cos 4 + 1/4

Answer 2

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