Question

Which of the following is/are TRUE ?

• A. \(∫^1_0 ∫^1_0 𝑒^{ max}(𝑥 ^2 , 𝑦^ 2) 𝑑𝑥\,\, 𝑑𝑦 = 𝑒 − 1\)
• B. \(∫^1_0 ∫^1_0 𝑒^{ min}(𝑥 ^2 , 𝑦^ 2) 𝑑𝑥\,\, 𝑑𝑦 = ∫ ^1_0𝑒^{ 𝑡^ 2} 𝑑𝑡 − ( 𝑒 − 1)\)
• C. \(∫^1_0 ∫^1_0 𝑒^{ max}(𝑥 ^2 , 𝑦^ 2) 𝑑𝑥\,\, 𝑑𝑦 = 2∫ ^1_0∫ ^1_y𝑒^{ 𝑡^ 2} \,dx\,dy\)
• D. \(∫^1_0 ∫^1_0 𝑒^{ max}(𝑥 ^2 , 𝑦^ 2) 𝑑𝑥\,\, 𝑑𝑦 = 2∫ ^1_0∫ ^y_1𝑒^{ y^ 2} \,dx\,dy\)

Answer 1

The Correct Option is A, C

To solve the given problem, we need to evaluate and verify the integrals and options presented. The problem involves evaluating double integrals where the integrand is the exponential function with either the maximum or minimum of two variables, \(x^2\) and \(y^2\), as its exponent.

1.  First, consider the option: \(∫^1_0 ∫^1_0 e^{\max(x^2, y^2)} \, dx \, dy = e - 1\).
   • The integrand \(e^{\max(x^2, y^2)}\) switches values depending on whether \(x^2\) or \(y^2\) is larger in the unit square defined by \(0 \leq x, y \leq 1\).
   • We can split the integral into two regions: where \(x^2 \geq y^2\) and \(y^2 > x^2\).
   • On the line \(y = x\), \(x^2 = y^2\). The main region of interest where \(x^2 \geq y^2\) is below this line, and where \(y^2 > x^2\) is above.
   • Evaluate:
     \(2∫^1_0 ∫^x_0 e^{x^2} \, dy \, dx + 2∫^1_0 ∫^y_0 e^{y^2} \, dx \, dy\)
     This simplifies to two regions of identical integration.
   • Solving this integral gives us the solution \(e - 1\). Thus, this is a correct statement.
2. Now consider the option: \(∫^1_0 ∫^1_0 e^{\max(x^2, y^2)} \, dx \, dy = 2∫^1_0 ∫^1_y e^{t^2} \, dx \, dy\).
   • The integral on the right splits the domains similarly by flipping the integration bounds when \(y\) is higher.
   • Rearranging and evaluating both sides of the integral, we find they are equivalent.
   • Hence, this statement is also true.
3. For the other statements, either the breakdown steps don't match or the integration boundaries in equations don't lead to equivalency.
   • Specifically, the integration ranges are critically different, leading to non-compatible boundary evaluations.

In conclusion, the statements:
\(∫^1_0 ∫^1_0 e^{\max(x^2, y^2)} \, dx \, dy = e - 1\) and \(∫^1_0 ∫^1_0 e^{\max(x^2, y^2)} \, dx \, dy = 2∫^1_0 ∫^1_y e^{t^2} \, dx \, dy\) are true.