Find the centroid of each region. The centroid is the center of mass of a region with constant density. The region bounded by y=x, y=-x and x=4

A lamina with constant density p(x, y) = p occupies the given region. Find the moments of inertia Ix and Iy and the radii of gyration ^=x and ^=y. The rectangle

$$0 ≤ x ≤ b, 0 ≤ y ≤ h$$

Find the exact arc length of the curve over the interval. $x=\frac{1}{8} y^4+\frac{1}{4} y^{-2}$ from $y=1$ to $y=4$

Use cylindrical shells to find the volume of the solid generated when the region enclosed by the given curves is revolved about the y-axis. y=1/x, y=0, x=1, x=3

Use cylindrical shells to find the volume of the solid generated when the region enclosed by the given curves is revolved about the x-axis. y=x², x=1, y=0

Use cylindrical shells to find the volume of the solid generated when the region enclosed by the given curves is revolved about the x-axis. y²=x, y=1, x=0

A lamina occupies the part of the disk

$$x^2+y^2≤a^2$$

that lies in the first quadrant. Find the center of mass of the lamina if the density function is

$$p(x,y)=xy^2$$

Use cylindrical shells to find the volume of the solid generated when the shaded region is revolved about the indicated axis.

A lamina occupies the part of the disk

$$x^2+y^2≤a^2$$

that lies in the first quadrant. Find the centroid of the lamina.

Use cylindrical shells to find the volume of the solid generated when the shaded region is revolved about the indicated axis.

Find the volume of the solid that results when the region enclosed by $y=\sqrt{x}$, $y=0$, and $x=9$ is revolved about the line $x=9$.

Find the volume of the solid that results when the region enclosed by $y=\sqrt{x} y=0$, and $x=9$ is revolved about the line $x=9$.

Find the volume of the solid that results when the region enclosed by $y=x^2$ and $y=x^3$ is revolved about the line $x=1$.

Integrate f over the given region.

$$f ( x , y ) = x ^ { 2 } + y ^ { 2 }$$

over the triangular region with vertices (0, 0), (1, 0), and (0, 1)