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.

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.

Substitute the cylindrical coordinate expressions into the integral. The integrand becomes r², and the volume element is r dz dr dθ. The limits of integration are 0 to 2π for θ, 0 to 1 for r, and 0 to 1-r² for z.

Substitute the cylindrical coordinate expressions into the integral. The integrand becomes r², and the volume element is r dz dr dθ. The limits of integration are 0 to 2π for θ, 0 to 1 for r, and 0 to 1-r² for z.

Next, integrate the result from the previous step with respect to r. Expand the term r³(1-r²) to r³ - r^5 and then integrate term by term. Evaluate the definite integral from 0 to 1.

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.

Evaluate ∬∫∫_E (x² + y²) dV where E is the region bounded by $z = 1 - x² - y²$ and the xy-plane.

Answer: The value of the integral is $\frac{\pi}{6}$ .

Step-by-step solution

Step 1: Identify the region of integration

The region $E$ is bounded above by the paraboloid $z = 1 - x^2 - y^2$ and below by the $xy$ -plane ( $z=0$ ). The intersection of the paraboloid with the $xy$ -plane is $0 = 1 - x^2 - y^2$ , which simplifies to $x^2 + y^2 = 1$ . 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 $(x^2 + y^2)$ , it is convenient to convert to cylindrical coordinates. In cylindrical coordinates, $x^2 + y^2 = r^2$ . The bounds for $z$ become $0 \le z \le 1 - r^2$ . The disk in the $xy$ -plane is described by $0 \le r \le 1$ and $0 \le \theta \le 2\pi$ .

Step 3: Set up the integral in cylindrical coordinates

Substitute the cylindrical coordinate expressions into the integral. The integrand becomes $r^2$ , and the volume element is $r \, dz \, dr \, d\theta$ . The limits of integration are $0$ to $2\pi$ for $\theta$ , $0$ to $1$ for $r$ , and $0$ to $1-r^2$ for $z$ .

Step 4: Evaluate the innermost integral with respect to z

First, integrate with respect to $z$ . Treat $r^3$ as a constant during this integration. The result is $r^3$ multiplied by the upper limit of $z$ , which is $1-r^2$ .

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 $r^3(1-r^2)$ to $r^3 - r^5$ 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 $\theta$ . The integral of a constant over an interval is the constant multiplied by the length of the interval.

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Evaluate ∬∫∫_E (x² + y²) dV where E is the region bounded by z=1−x2−y2z = 1 - x² - y²z=1−x2−y2 and the xy-plane.