Compute

1) 15

2) 16

3) 17

4) 18

Question

Compute

1) 15

2) 16

3) 17

4) 18

BlackTom AI · Accepted Answer

Integrate with respect to y first, since the integrand is given as an iterated integral ∫_{x=0}^{2} ∫_{y=0}^{3} (3x^2 − y) dy dx.

1) Relevant concepts
- Evaluation of iterated (double) integrals by successive single-variable integration.
- Constants with respect to the inner variable can be treated as constants during that inner integration.

2) Step-by-step reasoning and intermediate calculations
- Inner integral (with respect to y):
  ∫_{y=0}^{3} (3x^2 − y) dy
  = ∫_{0}^{3} 3x^2 dy − ∫_{0}^{3} y dy
  = 3x^2 · (y)|_{0}^{3} − (y^2/2)|_{0}^{3}
  = 3x^2 · 3 − (9/2)
  = 9x^2 − 9/2.

- Now integrate this result with respect to x from 0 to 2:
  ∫_{x=0}^{2} (9x^2 − 9/2) dx
  = 9 ∫_{0}^{2} x^2 dx − (9/2) ∫_{0}^{2} 1 dx
  = 9 · (x^3/3)|_{0}^{2} − (9/2) · (x)|_{0}^{2}
  = 9 · (8/3) − (9/2) · 2
  = 24 − 9
  = 15.

3) Why the correct answer is correct
The arithmetic follows directly from standard antiderivatives: ∫x^2 = x^3/3 and ∫y = y^2/2. Treating 3x^2 as constant during the y-integration and then integrating the resulting polynomial in x gives the value 15.

4) Why the other options are incorrect
Options 2 (16), 3 (17), and 4 (18) do not equal the computed exact value 15; they result from arithmetic mistakes or incorrect antiderivatives. The detailed stepwise calculation above rules them out.

Conclusion: The value of the integral is 15, so the correct choice is option 1, matching the provided answer.

Compute 1) 15 2) 16 3) 17 4) 18 Single choice

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