Question

Let E be the solid bounded by the planes

$z=x+y$, $z=0$, $y=0$, $y=x$, $x=3$.

Set up all six orders of integration for the evaluation of

$\int \int \int _{E}f(x,y,z) dV$

as an iterated integral.

Answer 1

Data from the given question z = x + y z = 0, y = 0 y = x, x = 3 I = tripleintegral F (x, y, z) dV I = integral^3_x = 0 integral^x_y = 0 integral^z = x + y_z = 0 dV Let us assume f (x, y, z) = e^x (y + z) Then, I = integral^3_x = 0 integral^x_y = 0 integral^x + y_3 = 0 e^x (y + z) dz dy dx I = integral^3_x = 0 e^x integral^x_y = 0 [y (x + y) + (x + y)^2/2] dy dx = integral^3_x = 0 e^x integral^x_y = 0 [2 y (x + y) + (x + y)^2/0] dy dx = integral^3_x = 0 e^x integral^x_y = 0 [2 xy + 2 y^2 + x^2 + y^2 + 2 xy/2] dy dx \r\n I = integral^3_x = 0 e^x/2 integral^x_y = 0 (2 xy + 2 y^2 + x^2 + y^2 + 2 xy) dy dx