Respuesta :
Answer:
0
Step-by-step explanation:
From your problem, we have to extract the information that are important from the first two intregrals so we can solve the double integral.
[tex]\int\limits^{-1}_{-4} {f(x)} \, dx = 0 [/tex]
We also have that:
[tex] \int\limits^{3}_{1} {g(x)} \, dx = 3 [/tex]
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With this, now we can solve the double integral.
Since the limits of integration are constant, i can use dA both as dydx or dxdy. I am going to use dydx.
So the double integral will be:
[tex] \int \limits^{-1}_{-4} \int \limits^{3}_{1} {g(y)} {f(x)} dy dx\ [/tex]
We solve a double integral from the inside to the outside, so the first integral we solve is:
[tex] \int \limits^{3}_{1} \y g(y) \x f(x) dy [/tex]
f is a function of x and we are integrating dy, so this means that f is a constant. Our integral now is this:
[tex] \x f(x) \int \limits^{3}_{1} \y g(y) dy [/tex]
From above, we have that
[tex]\int \limits^{3}_{1} \x g(x) dx = 3[/tex]
So,
[tex]\int \limits^{3}_{1} \y g(y) dy = 3[/tex]
Now we have to solve the outside integral:
[tex] 3\int\limits^{-1}_{-4} {f(x)} \, dx[/tex]
We know that
[tex] \int\limits^{-1}_{-4} {f(x)} \, dx = 0[/tex]
So the double integral will be 0
