Final answer:
To find the volume of the given solid bounded by the parabolic cylinder y=x² and the planes z=0, z=6, y=4, we can use a triple integral. The volume is found by integrating the function 1 over the given region. The volume of the solid is 128 cubic units.
Step-by-step explanation:
To find the volume of the given solid, we can use a triple integral. The solid is bounded by the parabolic cylinder y=x² and the planes z=0, z=6, y=4. We need to integrate the function 1 with respect to x, y, and z over the given region to find the volume.
Step 1: Set up the limits of integration:
- For x: 0 to 4
- For y: x² to 4
- For z: 0 to 6
Step 2: Set up the triple integral:
V = ∫∫∫ 1 dz dy dx
Step 3: Evaluate the triple integral:
V = ∫₀⁶ ∫ₓ²⁴ ∫₀⁴ 1 dz dy dx
Step 4: Simplify the integral:
V = ∫₀⁴ ∫ₓ²⁴ 6 dx dy
Step 5: Evaluate the double integral:
V = ∫₀⁴ 6(x² - 0) dx = 6∫₀⁴ x² dx
Step 6: Evaluate the remaining integral:
V = 6[(x³/3)|₀⁴] = 6[(4³/3) - (0³/3)] = 6(64/3) = 128
Therefore, the volume of the given solid is 128 cubic units.