Question12Question 12Numerical Entry Question This is an example of a Numerical Entry question, where you will be required to provide a numerical answer. Integrate the function [math: f(x)=x] with respect to [math: x] between the limits [math: X1=2] and [math: X2=4]. [math: ∫24x dx=] [input] .
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BlackTom AI · Accepted Answer

The antiderivative of x is x^2/2, so use the Fundamental Theorem of Calculus to evaluate the definite integral.

1) Relevant knowledge points and concepts
- Antiderivative (indefinite integral): ∫ x dx = x^2/2 + C.
- Fundamental Theorem of Calculus: ∫_{a}^{b} f(x) dx = F(b) − F(a), where F is an antiderivative of f.
- Geometric interpretation: area under y = x between x = 2 and x = 4 is the area of a trapezoid (or difference of two right triangles).

2) Step-by-step reasoning
- Find an antiderivative F(x) of f(x) = x: F(x) = x^2/2.
- Evaluate F at the upper limit 4 and at the lower limit 2.
- Subtract: F(4) − F(2).

3) Calculations with intermediate steps
- F(4) = (4^2)/2 = 16/2 = 8.
- F(2) = (2^2)/2 = 4/2 = 2.
- ∫_{2}^{4} x dx = F(4) − F(2) = 8 − 2 = 6.

(Geometric check) Area method: average height = (f(2)+f(4))/2 = (2+4)/2 = 3; width = 4−2 = 2; area = average height × width = 3×2 = 6.

4) Why the correct answer is correct
The calculation follows the Fundamental Theorem exactly and matches the geometric area under the straight line y = x over [2,4], so the value 6 is correct.

5) Common incorrect answers and why they are wrong
- 12: arises if someone computes F(x)=x^2 (forgetting the 1/2 factor) so 16−4 = 12 (incorrect antiderivative).
- 8: might come from using only F(4) = 8 and forgetting to subtract F(2).
- 4: could be from arithmetic errors or misinterpreting width as 1 instead of 2.

6) Conclusion
The definite integral ∫_{2}^{4} x dx equals 6, which matches the provided answer.

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