1. A student evaluates $\int_5^2 (4x)\,dx$ and obtains $42$. A peer claims the answer should be $-42$. Which student is correct, and why?
- A.The first student is correct; the integral equals $42$ because area is always positive.
- B.The peer is correct; $\int_5^2 (4x)\,dx = -\int_2^5 (4x)\,dx = -42$.
- C.Neither is correct; the integral is $0$ because the limits encompass equal positive and negative areas.
- D.The integral is undefined because the lower limit is greater than the upper limit.
View Answer
Answer: The peer is correct; $\int_5^2 (4x)\,dx = -\int_2^5 (4x)\,dx = -42$.
Verify $\int_2^5 4x\,dx$: $\int_2^5 4x\,dx = [2x^2]_2^5 = 2(25) - 2(4) = 50 - 8 = 42$. Apply the reversal property: $\int_5^2 4x\,dx = -\int_2^5 4x\,dx = -42$. The peer is correct. Why distractors fail: Option A ignores the reversal-of-limits property. Option C has no basis. Option D is wrong because the definite integral is well-defined even when the lower limit exceeds the upper limit.