Explanation:
We are given the solid E bounded by z=0, x=0, z=y−5x and y=15.
To express the integral ∭ E f(x,y,z)dV as an iterated integral in six different ways, we can use the bounds of integration for each variable x, y, and z.
Method 1: Integrating with respect to z first
The bounds of z are from 0 to y-5x. The bounds of y are from x+5x to 15. The bounds of x are from 0 to 3.
∫0^3 ∫x+5x^15 ∫0^(y-5x) f(x,y,z) dz dy dx
Method 2: Integrating with respect to z last
The bounds of z are from 0 to y-5x. The bounds of y are from x+5x to 15. The bounds of x are from 0 to 3.
∫x=0^3 ∫y=x+5x^15 ∫z=0^(y-5x) f(x,y,z) dz dy dx
Method 3: Integrating with respect to y first
The bounds of y are from 5x to 15. The bounds of x are from 0 to (y/5).
∫0^3 ∫5x^15 ∫0^(y-5x) f(x,y,z) dz dy dx
Method 4: Integrating with respect to y last
The bounds of y are from 5x to 15. The bounds of x are from 0 to (y/5).
∫5x^15 ∫0^(y/5) ∫z=0^(y-5x) f(x,y,z) dz dx dy
Method 5: Integrating with respect to x first
The bounds of x are from 0 to (y/5). The bounds of y are from 5x to 15. The bounds of z are from 0 to y-5x.
∫0^3 ∫5x^15 ∫0^(y-5x) f(x,y,z) dz dy dx
Method 6: Integrating with respect to x last
The bounds of x are from 0 to (y/5). The bounds of y are from 5x to 15. The bounds of z are from 0 to y-5