Find the volume of the solid formed when the shaded region

Question

1. Find the volume of the solid formed when the shaded region is revolved about the indicated axis.

2. Find the volume of the solid generated by revolving the region bounded by y = x (1 +x³)^1/4 over 0 ≤ x ≤ 1 about the x-axis.

3. A city's water storage tank is in the shape of the solid of revolution generated by revolving the region under the graph of f(x)= 50 √(1 - x²/1600) from x = -40 feet to x = 40 feet about the x-axis.

(a) What is the volume of this tank?

(b) It is known that 1 ft³ hold 7.48 gallons. How much water does this tank hold?

4. Find the volume of the solid that results when the region enclosed by y = sin(x), y = cos(x), x = 0, x = π/4 is revolved about the x-axis. [Hint: Use a trigonometric identity to simplify the integrand.]

5. Let V be the volume of the solid that results when the region enclosed by y = 1/x, y = 0, x = b, and x = 2 (0 < b < 2) is revolved about the x-axis. Find the value of b for which V = 3.