Exam 12: Multiple Integrals

Find the moment of inertia $I _ { x }$ about the x-axis and the moment of inertia $I _ { y }$ about the y-axis for the region in the first quadrant bounded by y = x and $y ^ { 2 } = x ^ { 3 }$ , assuming p = 1.

Find the y-coordinate of the centroid of the semiannular plane region given by $1 \leq x ^ { 2 } + y ^ { 2 } \leq 4$ , $y \geq 0$ . Sketch the plane region and plot the centroid in the graph.

Use cylindrical coordinates to find $\iiint _ { E } z d V$ , where R is the region bounded by $z = \sqrt { x ^ { 2 } + y ^ { 2 } }$ and $z = x ^ { 2 } + y ^ { 2 }$ .

Evaluate $\iiint _ { E } z ^ { 2 } d V$ , where E is the solid bounded by the ellipsoid $\frac { x ^ { 2 } } { 9 } + \frac { y ^ { 2 } } { 25 } + z ^ { 2 } = 1$ .

Find the volume of the solid in the first octant that is bounded by the plane y + z = 4, the cylinder $y = x ^ { 2 }$ , and the xy- and yz-planes.

Find a transformation x = x (u, v), y = y (u, v) maps the region in the uv-plane into the xy-plane.

Give a geometric description of the solid S whose volume in spherical coordinates is given by $V = \int _ { 0 } ^ { 2 x } \int _ { x / 4 } ^ { x / 2 } \int _ { 0 } ^ { 2 } \rho ^ { 2 } \sin \phi d \rho d \phi d \theta$ .

Evaluate the triple integral $\iiint _ { E } y d V$ , where E is the solid bounded by the coordinate planes and the plane 2x + y + z = 4.

Evaluate $\iint _ { R } \sin \left( 9 x ^ { 2 } + 4 y ^ { 2 } \right) d A$ by making an appropriate change of variables, where R is the region in the first quadrant bounded by the ellipse $9 x ^ { 2 } + 4 y ^ { 2 } = 1$ .

Evaluate the double integral $\iint _ { R } x d A$ , where $R = \left\{ ( x , y ) \mid 0 \leq x \leq 2 , \frac { 1 } { 2 } x \leq y \leq 3 - x \right\}$ .

Rewrite $\iint _ { R } f ( x , y ) d A$ as an iterated integral with y as the variable of integration in the outer integral, where R is the region shown below.

Rewrite the integral $\int _ { 0 } ^ { 1 } \int _ { x } ^ { 1 + \sqrt { 1 - x ^ { 2 } } } x d y d x$ in terms of polar coordinates, then evaluate the integral.

Find the Jacobian of the transformation $x = u - v ^ { 2 }$ , $y = u + v ^ { 2 }$ .

Evaluate the triple integral $\iiint _ { E } \sqrt { x ^ { 2 } + y ^ { 2 } + z ^ { 2 } } d V$ in spherical coordinates, where E is the solid bounded by the hemisphere $z = \sqrt { 4 - x ^ { 2 } - y ^ { 2 } }$ and the plane z = 0.

Evaluate $\iiint _ { z } e ^ { \left( x ^ { 2 } + y ^ { 2 } + z ^ { 2 } \right) ^ { 3 / 2 } } d V$ , where E is the solid bounded by the sphere $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 1$ and the cone $z = \frac { 1 } { \sqrt { 3 } } \sqrt { x ^ { 2 } + y ^ { 2 } }$ .

Evaluate the iterated integral $\int _ { 0 } ^ { \sqrt { 2 } / 2 } \int _ { x } ^ { \sqrt { 1 - x ^ { 2 } } } \sin \left( \pi \left( x ^ { 2 } + y ^ { 2 } \right) \right) d y d x$ by converting to polar coordinates.

Find the y-coordinate of the center of mass of the lamina that occupies the region $D = \{ ( x , y ) \mid 0 \leq x \leq 1,0 \leq y \leq 1 \}$ and whose density function at any point is the distance from that point to the y-axis.

Write $\iint _ { R } f ( x , y ) d A$ as an iterated integral in polar coordinates, where R is the region shown below.

Find a transformation x = x (u, v), y = y (u, v) maps the region in the uv-plane into the xy-plane.

Evaluate the iterated integral $\int _ { 0 } ^ { x } \int _ { 0 } ^ { 1 } x \sin y d x d y$ .