Integral, volume

1. Evaluate the integral, given that 𝑎 > 1. 𝑎 𝑥 2 ∫ ∫(𝑒 𝑥 + 𝑒 𝑥𝑦 ) 𝑑𝑦𝑑𝑥 1 0 2. Evaluate the integral, below. 0.4459𝜋 sin (𝑦) ∫ ∫ 0 0.5 𝑦 2 𝑦 𝑑𝑥 𝑑𝑦 3. In the previous problem, describe in geometric terms what the number you just found represents. 4. Evaluate ∬𝐷 (2𝑥 − 𝑦) 𝑑𝐴 where D is bounded by the circle, centered at the origin with radius2. In problems 5 and 6, find the volume of the solid described below: 5. The tetrahedron enclosed by the coordinate planes and the plane 2𝑥 + 𝑦 + 𝑧 = 4. 6. Under the surface 𝑧 = 𝑥 2 𝑦 and above the portion of the semi-circle centered at the origin, radius 5 that lies in quadrants 1 and 2. 7. Set up the double integration that would yield the volume bound by the planes 𝑧 = 0 and 𝑧 = 14 + 2𝑥 − 3𝑦 and by the curves in the plane 𝑧 = 0, 𝑥 − 𝑦 2 + 2 = 0 and 𝑥 2 + 𝑦 − 10 = 0. You do not have to compute the integrals. You may use a calculator to estimate the numeric boundaries of the outside integral. . 8. Evaluate ∬𝐷 sin (𝑥 2 + 𝑦 2 ) 𝑑𝐴 where D is the region in the first quadrant that lies between the circles with center at the origin and radii 1 and 3, respectively. 𝑦2 9. Evaluate ∬𝐷 ( 2 𝑥 +𝑦 2 ) 𝑑𝐴 where D is the region that lies between the circles given by 𝑥 2 + 𝑦 2 = 𝑎2 and 𝑥 2 + 𝑦 2 = 𝑏 2 with 0 < 𝑎 < 𝑏. In problems 10-13, set up the integrations that will yield the desired results. You can use an online calculator to evaluate the integrals. 10. Find the mass and center of mass of the lamina that occupies the region D if 𝐷 = { (𝑥, 𝑦) ∶ 1 ≤ 𝑥 ≤ 3, 1 ≤ 𝑦 ≤ 4 }; 𝜌(𝑥, 𝑦) = 𝑘𝑦 2 11. Find the mass and center of mass of the lamina that occupies the triangular region inside the triangle whose vertices are (0, 0), (0,1), and (2,3). 𝜌(𝑥, 𝑦) = 𝑥. 12. Find the mass and center of mass of the part of the disk 𝑥 2 + 𝑦 2 ≤ 1 that lies in the first quadrant if the density at any point is proportional to the distance from the origin. 13. A lamina formed by the part of the region bound by 𝑦 = sin (𝑥) and 𝑥 = 0 on the domain interval 𝑥 ∈ [ 0, 𝜋 ] has constant density given by 𝜌. Find the moments of inertia about each axis, 𝐼𝑥 and 𝐼𝑦 and the radii of gyration, 𝑥̿ and 𝑦̿. 14. Use Lagrange multipliers to find the minimal and maximal values of the function given by 𝑓 = 𝑥 2 + 𝑦 2 + 𝑧 2 under the constraint 𝑥 4 + 𝑦 4 + 𝑧 4 = 1