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Use a triple integral to find the volume of the given solid. The solid bounded by the parabolic cylinder ( y=x² ) and the planes ( z=0, z=6, y=4 ).

User Ashrugger
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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.

User Finico
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