Evaluate the triple integral ∫∫∫_E (z)dV where E is the solid bounded by the cylinder y^2+z^2=1225 and the planes x=0, y=7x and z=0 in the first octant.

To evaluate the given triple integral, let's first understand the region E that is being described.

The cylinder y^2+z^2=1225 represents a circular cross-section centered at the origin (0,0,0) in the y-z plane. Since it is bounded in the first octant, we know that x, y, and z are all non-negative.

The planes x=0 and y=7x define the boundaries of E in the x-y plane. Since x cannot be negative, we know that E is bounded by the y-axis, and y=7x is the upper limit for y.

The plane z=0 defines the bottom boundary of E. Since z cannot be negative, we know that E is bounded by the xy-plane.

So visually, E is a solid in the first octant that consists of a circular cross-section in the y-z plane, with the boundaries defined by x=0, y=7x, and z=0.

To evaluate the triple integral ∫∫∫_E (z)dV, we need to determine the limits of integration for each variable. In this case, since E is described in terms of x, y, and z, we will integrate with respect to x, y, and z.

Let's start with integrating with respect to x. The lower limit of x is 0, as given by the plane x=0. The upper limit for x is found by the equation y=7x. Solving for x, we have x=y/7.

Next, let's integrate with respect to y. The lower limit of y is 0 since E is bounded by the y-axis. The upper limit for y is given by the equation y=7x, where x=y/7.

Finally, let's integrate with respect to z. The lower limit of z is 0 as given by the plane z=0. The upper limit of z is determined by the cylinder equation y^2+z^2=1225. Solving for z, we have z=sqrt(1225-y^2).

Now, we have the limits of integration for each variable:

∫∫∫_E (z)dV = ∫[0 to √1225]∫[0 to 7x]∫[0 to sqrt(1225-y^2)] (z) dz dy dx

To evaluate this triple integral, we can apply the rules of integration and evaluate it step by step.