Exam 16: Multiple Integration

Find the center of mass of a thin plate covering the given region with the given density function. -The region bounded below by the parabola and above by the line , with density

Write an equivalent double integral with the order of integration reversed. -

Find the center of mass of a thin plate of constant density covering the given region. -The region bounded by y = 6 - x and the axes

Write an equivalent double integral with the order of integration reversed. -

Solve the problem. -Find the center of mass of the region of constant density bounded by the paraboloid and the xy-plane.

Use the given transformation to evaluate the integral. - where R is the parallelogram bounded by the lines

Find the Jacobian for the given transformation -

Solve the problem. -Let D be the smaller cap cut from a solid ball of radius 7 units by a plane 4 units from the center of the sphere. Set up the triple integral for the volume of D in cylindrical coordinates.

Evaluate the integral by changing the order of integration in an appropriate way. -

Evaluate the spherical coordinate integral. -

Solve the problem. -If f(x, y) = ( 3000 )/(1 + /2) represents the population density of a planar region on Earth, where x and y are measured in miles, find the number of people within the rectangle -7 $\le$ x $\le$ 7 and -3 $\le$ y $\le$ 0.

Find the volume of the indicated region. -the region bounded by the paraboloid and the cylinder

Use the given transformation to evaluate the integral. - where R is the interior of the ellipsoid

Evaluate the integral. -

Evaluate the double integral over the given region. - R = {(x, y): 9 $\le$ x $\le$ 10, 1 $\le$ y $\le$ 5}

Find the area of the region specified in polar coordinates. -the region inside and outside

Integrate the function f over the given region. -f(x, y) = + over the trapezoidal region bounded by the x-axis, y-axis, line and line

Find the center of mass of a thin plate of constant density covering the given region. -The region between the curve y = and the x-axis from to

Integrate the function f over the given region. -f(x, y) = over the region bounded by the x-axis, line and curve