Tính ∭E108y−72xdV
với E
là khối có biên là các mặt y=10−2z,y=0,z=2x,z=5
và x=0
.

Chú ý kết quả điền và...

Question

Tính ∭E108y−72xdV
 với E
 là khối có biên là các  mặt y=10−2z,y=0,z=2x,z=5
 và x=0
.

Chú ý kết quả điền vào là số thập phân làm tròn đến 3 chữ số sau dấu phẩy.

CramX · Accepted Answer

Step 1: This is a triple integral problem in the field of calculus.. The given integral is over a region E defined by the boundaries y=10−2z, y=0, z=2x, z=5, and x=0. The function to be integrated is 108y−72x. The goal is to evaluate the triple integral. Step 2: : Identify the order of integration. The order of integration is determined by the limits of the integral. In this case, the boundaries suggest that it is most convenient to integrate in the order dz, dy, dx. Step 3: : Set up the integral. The triple integral over the region E is given by: Step 4: : Determine the limits of integration. From the given boundaries, we can determine the limits of integration: - For z, from the equations y=10−2z and z=2x, we can see that z ranges from 2x to 5. - For y, from the equations y=10−2z and y=0, we can see that y ranges from 0 to 10-2z. - For x, from the equation x=0 and the fact that x is always non-negative, we can see that x ranges from 0 to 5/2. Step 5: : Substitute the limits into the integral. Substituting the limits into the integral, we get: Step 6: : Evaluate the integral. First, we integrate with respect to z: This simplifies to: Next, we integrate with respect to y: This simplifies to: Finally, we integrate with respect to x: This simplifies to: Simplifying further, we get: Step 7: : Evaluate the final result. Substitute z=5 into the final result: Final Answer: The value of the triple integral is -562.5. However, the problem asks for the answer to be rounded to three decimal places.

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Step 1:
This is a triple integral problem in the field of calculus.

Step 2:
: Identify the order of integration

Step 3:
: Set up the integral

Final Answer

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