Final answer:
To reverse the order of integration of ∫₀¹∫ₓ¹cos(y²) dy dx, we change the limits accordingly, resulting in ∫₀¹∫₀ₓcos(y²) dx dy, and then evaluate the inner integral, ending up with ∫₀¹ycos(y²) dy, which may require special functions or numerical methods.
Step-by-step explanation:
The question asks to reverse the order of integration and evaluate the integral ∫₀¹∫ₓ¹cos(y²) dy dx. To reverse the order of integration, we need to consider the limits of both variables x and y. The original region of integration has x varying from 0 to 1, and for each x, y varies from x to 1.
Reversing the Order of Integration
When we reverse the order, y will be the outer variable and will vary from 0 to 1, but the inner variable x will vary from 0 to y. So, the new double integral would be: ∫₀¹∫₀ₓcos(y²) dx dy.
Evaluating the Reversed Integral
Now evaluating the inner integral with respect to x is straightforward since the integrand does not contain x. It becomes: ∫₀¹cos(y²) · (y - 0) dy, which simplifies to ∫₀¹ycos(y²) dy. To evaluate this integral, one may need to use special functions or numerical methods since the integral of cos(y²) is not elementary.