Triple Integrals:

1. Evaluate  \(\displaystyle{ \int_0^1 \int_1^2\int_2^3 (x^2+\ln y+z)\,dx \, dy\, dz} \)    solution

2. Evaluate  \(\displaystyle{ \iiint_B 4x^2yz^3 \, dV} \), where \( B= [0, 2]\times [0,1] \times [1,2] \).   solution

3. Evaluate  \(\displaystyle{ \iiint_E 3z \, dV} \), where \( E \) is the region  bounded by the planes  \(2x+y+z=2\) and the coordinate planes.   solution

4. Set up (do not evaluate) as an iterated integral  \(\displaystyle{ \iiint_E 20y \, dV} \), where \( E \) is the region above  the triangle with vertices \((0, 0), \, (1, 0)\), and \((1, 1)\) in the \(xy\)-plane, and below the graph of \(z = x^2 + y^2\).   solution

5. Evaluate  \(\displaystyle{ \iiint_E \sin y \, dV} \), where \( E \) is the region below the plane \(z=x\) and above the triangular region with vertices \((0, 0, 0), \, (\pi, 0, 0)\), and \((0, \pi, 0)\).   solution

6. Evaluate  \(\displaystyle{ \iiint_T 3y^2 \, dV} \), where \( T \) is the solid tetrahedron with vertices \((0, 0, 0), \, (2, 0, 0), \, (0, 2, 0)\),  and \((0, 0, 2)\).   solution

7. Set up (but do not evaluate) a triple integral whose value is the volume of the region under the plane given by \(x+2y-z=0\) and above the region in the xy-plane bounded by the graphs of \(y=3x\) and \(y=x^2-x\).    solution