Finding the Area of a Region on a Plane Bounded by a Cylinder: A Detailed Guide

Understanding how to calculate the area of regions on a plane that are bounded by a cylinder is a fundamental concept in multivariable calculus. In this article, we will walk through the detailed process of finding the area of the region on the plane x - 2y - z 1 that is bounded by the cylinder x^2 y^2 4.

Expressing the Plane Equation

To start, let's express z in terms of x and y. The equation of the plane is:

x - 2y - z 1

Rearranging it gives:

z x - 2y - 1

Identifying the Projection on the xy-Plane

The cylinder x^2 y^2 4 describes a circular region of radius 2 in the xy-plane. The region of interest on the plane is the projection of this circular region onto the plane.

Setting Up the Double Integral for the Area

The area A of the region on the plane can be calculated using the formula:

A iint_R sqrt{1 left(frac{partial z}{partial x}right)^2 left(frac{partial z}{partial y}right)^2} dA

where R is the projection of the region onto the xy-plane and dA is the area element in the xy-plane.

Calculating the Partial Derivatives

Next, we calculate the partial derivatives of z with respect to x and y.

frac{partial z}{partial x} 1

frac{partial z}{partial y} -2

Substituting into the Area Formula

Substituting the partial derivatives into the area formula gives:

A iint_R sqrt{1 1^2 (-2)^2} dA iint_R sqrt{1 1 4} dA iint_R sqrt{6} dA

Calculating the Area of the Region R

The area of the circular region R defined by x^2 y^2 leq 4 is:

text{Area of } R pi cdot 2^2 4pi

Final Area Calculation

Finally, we can find the area A:

A sqrt{6} cdottext{Area of } R sqrt{6} cdot 4pi 4pisqrt{6}

Thus, the area of the region on the plane x - 2y - z 1 that is bounded by the cylinder x^2 y^2 4 is:

boxed{4pisqrt{6}}