Evaluate the double integral (a) int^(3_-1) int^(2_0)  x^(3) y dx dy

Expert Answers
sciencesolve eNotes educator| Certified Educator

You need to identify the inner integral such that:

`int_0^2 x^3 y dx`

Hence, you should evaluate this integral considering y as a constant such that:

`int_0^2 x^3 y dx = (y*x^4)/4|_0^2`

`int_0^2 x^3 y dx = y(2^4/4 - 0^4/4)`

`int_0^2 x^3 y dx = 4y`

You need to evaluate the outer integral, hence, you need to substitute 4y for inner integral such that:

`int_(-1)^3 int_0^2 x^3 y dx dy = int_(-1)^3 4y dy`

`int_(-1)^3 4y dy = 4y^2/2|_(-1)^3`

`int_(-1)^3 4y dy = 2(3^2 - (-1)^2)`

`int_(-1)^3 4y dy = 2(9 - 1) => int_(-1)^3 4y dy = 16`

Hence, evaluating the double integral `int_(-1)^3 int_0^2 x^3 y dx dy`  yields `int_(-1)^3 int_0^2 x^3 y dx dy = 16` .

lambert86 | Student

This is a question on double integral.

First do the integral w.r.t y assuming x is a constant.






Now integrate the function w.r.t x;



=2x^4 (2_0)


= 2*16


Therefore the answer is 32.