Evaluate ∬∫∫_E (x² + y²) dV where E is the region bounded by z=1 - x² - y² and the xy-plane.
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Step-by-Step Solution
Step 1: Identify the region of integration
The region E is bounded above by the paraboloid z=1−x2−y2 and below by the xy -plane (z=0). The intersection of the paraboloid with the xy -plane is 0=1−x2−y2, which simplifies to x2+y2=1. This means the region in the xy -plane is a disk of radius 1 centered at the origin.
Step 2: Convert to cylindrical coordinates
Due to the circular symmetry of the region and the integrand (x2+y2), it is convenient to convert to cylindrical coordinates. In cylindrical coordinates, x2+y2=r2. The bounds for z become 0≤z≤1−r2. The disk in the xy -plane is described by 0≤r≤1 and 0≤θ≤2π.
Step 3: Set up the integral in cylindrical coordinates
Substitute the cylindrical coordinate expressions into the integral. The integrand becomes r2, and the volume element is rdzdrdθ. The limits of integration are 0 to 2π for θ, 0 to 1 for r, and 0 to 1−r2 for z.
Step 4: Evaluate the innermost integral with respect to z
First, integrate with respect to z. Treat r3 as a constant during this integration. The result is r3 multiplied by the upper limit of z, which is 1−r2.
Step 5: Evaluate the middle integral with respect to r
Next, integrate the result from the previous step with respect to r. Expand the term r3(1−r2) to r3−r5 and then integrate term by term. Evaluate the definite integral from 0 to 1.
Step 6: Evaluate the outermost integral with respect to θ
Finally, integrate the constant result from the previous step with respect to θ. The integral of a constant over an interval is the constant multiplied by the length of the interval.