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Consider the region enclosed by the curve y = e^(x²) and the horizontal line y = e. Which expression does NOT correctly represent its area?

1. 2 * integral[1 to e] sqrt(logₑ y) dy
2. 2e - integral[-1 to 1] e^(x²) dx
3. integral[-1 to 1] (e - e^(x²)) dx
4. 2 * integral[0 to 1] sqrt(x) e^x dx

Correct answer: 2 * integral[0 to 1] sqrt(x) e^x dx

Solution

The area = integral[-1 to 1] (e - e^(x²)) dx = 2e - integral[-1 to 1] e^(x²) dx, and in terms of y it is 2*integral[1 to e] sqrt(ln y) dy. The option 2*integral[0 to 1] sqrt(x) e^x dx is not a valid representation.

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