Find the volume of the solid region bounded by the

Question

Solve the following problems, providing detailed steps wherever required:

1. Evaluate the iterated integral ₁∫^e₀∫¹ (x/x+y) dydx.

2. Let R = {(x, y): 0 ≤ x ≤ π, 0 ≤ y ≤ a}. Find the values of a, with 0 ≤ a ≤ π, for which ∫∫_Rsin(x + y) dA is equal to 1?

3. Sketch the region of integration and evaluate the integral ₀∫⁴-_√(16-y2)∫^√(16-y2)2xydxdy.

4. Evaluate the integral _-2∫²_x2∫^3-x2 xdydx.

5. Evaluate the integral ∫∫_R y² dA; R is bounded by x = 1, y = 2x + 2, and y = - x - 1.

6. Find the volume of the solid region for the solid S between the surfaces z = e^x-y and z = -e^x-y, where S intersects the xy plane in the region R = {(x, y): 0 ≤ x ≤ y, 0 ≤ y ≤ 1}.

7. Reverse the order of integration in the integral ₀∫²_x2∫^2xf(x, y)dydx.

8. Use a double integral to compute the area of the region bounded by the lines x = 0, x = 4, y = x, and y = 2x + 1. Also, make a sketch of the region.

9. Find the volume of the solid below the paraboloid z = 4 - x² - y² and above the region R = {(r, θ): 1 ≤ r ≤ 2, - π/2 ≤ θ ≤ π/2}.

10. Sketch the given region of integration R and evaluate the integral over R using polar coordinates

∫∫_R1/√(16-x²-y²) dA;

11. Sketch the region inside the lobe of the lemniscate r² = 2 sin2θ in the first quadrant. Also, express ∫∫_Rf(r, θ) dA as an iterated integral over R.

12. Sketch the region inside both the cardioid r = 1 + sin θ and the cardioid r = 1 + cos θ. Also, use integration to find its area.

13. Evaluate the integral ₀∫^π/2₁∫^∞(cosθ/r³)rdrdθ.

14. Evaluate the integral _-2∫²₁∫²₁∫^e(xy²/x)dzdxdy.

15. Evaluate the integral ₀∫^π₀∫^π₀∫^sinx sinydzdxdy.

16. Find the volume of the solid region bounded by the parabolic cylinder y = x² and the planes z = 3 - y and z = 0.

17. Find the average value in case of average temperature in the box D = {(x, y, z): 0 ≤ x ≤ ln2, 0 ≤ y ≤ ln4, 0 ≤ z ≤ ln8}, with a temperature distribution of T(x, y, z) = 100e^-x-y-z.

18. Two different tetrahedrons fill the region in the first octant bounded by the coordinate planes and the plane x + y + z = 4. Both solids have densities that vary in the z-direction between ρ = 4 and ρ = 8, according to the functions ρ₁ = 8 - z and ρ₂ = 4 + z. Find the mass of each solid.