Evaluate the surface integral. s z + x2y ds s is the part of the cylinder y2 + z2 = 16 that lies between the planes x = 0 and x = 3 in the first octant

The value of the surface integral which is part of the cylinder that lies between the planes x = 0 and x = 3 in the first octant is 192

The process of arriving at the above value is as follows:

Known parameters:

Question: The surface integral equation appears to have some parts left out as follows;

The equation of the cylinder is y² + z² = 16

The planes between which the cylinder lies = x = 0, and x = 3

The required parameter:

Evaluate the surface integral

Method:

We have;

z² = 16 - y²

∴ z = √(16 - y²)

Therefore;

z is a function of x, and y

z = g(x, y) = √(16 - y²)

g'ₓ = d(√(16 - y²))/dx = 0

The formula for surface integral is presented as follows;

Substituting gives;

Given that we have;

Therefore

The value of the surface integral = 192

Learn more about surface integral here:

link

Parameterize by

with and . The normal vector to has magnitude

so that the surface integral is

560

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