The total mass M is found by integrating the density function over the volume of the solid. We use cylindrical coordinates, so dV = r dz dr dθ. We integrate with respect to z first, then r, and finally θ. After evaluating the integrals, we find the total mass M = (64π)/(3).

The total mass M is found by integrating the density function over the volume of the solid. We use cylindrical coordinates, so dV = r dz dr dθ. We integrate with respect to z first, then r, and finally θ. After evaluating the integrals, we find the total mass M = (64π)/(3).

We integrate z² · r with respect to z, then r, and finally θ. The result of this integration is M_z = (64π)/(3).

The coordinates of the center of mass are given by x = (M_x)/(M), y = (M_y)/(M), and z = (M_z)/(M). Since M_x = 0 and M_y = 0, we have x = 0 and y = 0. Substituting the values for M_z and M, we find z = 1.

The coordinates of the center of mass are given by x = (M_x)/(M), y = (M_y)/(M), and z = (M_z)/(M). Since M_x = 0 and M_y = 0, we have x = 0 and y = 0. Substituting the values for M_z and M, we find z = 1.

Find the center of mass of the solid bounded by $z = 4 - x² - y²$ and $z = 0$ with density ρ(x,y,z) = z.

Answer: The center of mass is $(0, 0, 1)$ .

Step-by-step solution

Step 1: Identify the region and density function

The solid is bounded by the paraboloid $z = 4 - x^2 - y^2$ and the $xy$ -plane ( $z=0$ ). The density function is given as $\rho(x,y,z) = z$ . This means the density varies with height.

Step 2: Determine the limits of integration

The intersection of $z = 4 - x^2 - y^2$ and $z=0$ gives $4 - x^2 - y^2 = 0$ , or $x^2 + y^2 = 4$ . This is a circle of radius 2 in the $xy$ -plane. Converting to cylindrical coordinates, $x^2+y^2=r^2$ , so the region in the $xy$ -plane is $0 \le r \le 2$ and $0 \le \theta \le 2\pi$ . The $z$ limits are from $0$ to $4 - r^2$ .

Step 3: Calculate the total mass (M)

The total mass $M$ is found by integrating the density function over the volume of the solid. We use cylindrical coordinates, so $dV = r \, dz \, dr \, d\theta$ . We integrate with respect to $z$ first, then $r$ , and finally $\theta$ . After evaluating the integrals, we find the total mass $M = \frac{64\pi}{3}$ .

Step 4: Calculate the moments $M_x$ , $M_y$ , and $M_z$

Due to the symmetry of the solid and the density function with respect to the $z$ -axis, the center of mass will lie on the $z$ -axis. This means $M_x = 0$ and $M_y = 0$ , so $\bar{x} = 0$ and $\bar{y} = 0$ . We only need to calculate $M_z$ .

Step 5: Calculate $M_z$

We integrate $z^2 \cdot r$ with respect to $z$ , then $r$ , and finally $\theta$ . The result of this integration is $M_z = \frac{64\pi}{3}$ .

Step 6: Calculate the center of mass $(\bar{x}, \bar{y}, \bar{z})$

The coordinates of the center of mass are given by $\bar{x} = \frac{M_x}{M}$ , $\bar{y} = \frac{M_y}{M}$ , and $\bar{z} = \frac{M_z}{M}$ . Since $M_x = 0$ and $M_y = 0$ , we have $\bar{x} = 0$ and $\bar{y} = 0$ . Substituting the values for $M_z$ and $M$ , we find $\bar{z} = 1$ .

Solve your own maths question free →

Find the center of mass of the solid bounded by z=4−x2−y2z = 4 - x² - y²z=4−x2−y2 and z=0z = 0z=0 with density ρ(x,y,z) = z.