Final answer:
To evaluate the given integral, we integrate with respect to y first and then with respect to x, following the limits of integration.
Step-by-step explanation:
To evaluate the given integral, ∫₀¹∫ₓ¹ eʸ² d y d x, we need to integrate with respect to y first and then with respect to x.
The inner integral ∫ₓ¹ eʸ² d y can be evaluated as eʸ² evaluated at the upper limit x minus eʸ² evaluated at the lower limit 1.
So, the value of the inner integral is eʸ²(x) - eʸ²(1). Now, substitute the limits of integration to get eˣ² - e.
Then, evaluate the outer integral ∫₀¹ (eˣ² - e) d x. This integral can be evaluated as eˣ²x - ex evaluated at the upper limit 1 minus eˣ²x - ex evaluated at the lower limit 0.
Substituting the limits of integration gives the final result as e - 1.