We are given the double integral:

$\int_0^3 \int_{y/3}^1 y \cos(x^3 - 1) \, dx \, dy$

The problem asks to reverse the order of integration and calculate the result.

Step 1: Analyze the current limits

The current limits are as follows:

• For $y$, $0 \leq y \leq 3$,
• For $x$, $\frac{y}{3} \leq x \leq 1$.

Step 2: Find the new limits after reversing the order

To change the order of integration, we need to express the region in the $x$-$y$ plane. From the current limits:

• $x$ ranges from $\frac{y}{3}$ to 1,
• $y$ ranges from 0 to 3.

In terms of $x$, $y$ varies between:

• The line $y = 3x$, which comes from solving $x = \frac{y}{3}$,
• And $y = 3$.

Thus, the new integral limits are:

• For $x$, $0 \leq x \leq 1$,
• For $y$, $3x \leq y \leq 3$.

Step 3: Set up the reversed integral

The new order of integration will be:

$\int_0^1 \int_{3x}^3 y \cos(x^3 - 1) \, dy \, dx$

Step 4: Perform the integration

Now, let's compute the inner integral with respect to $y$:

$\int_{3x}^3 y \cos(x^3 - 1) \, dy$

Since $\cos(x^3 - 1)$ is independent of $y$, we can factor it out:

$\cos(x^3 - 1) \int_{3x}^3 y \, dy = \cos(x^3 - 1) \left[ \frac{y^2}{2} \right]_{3x}^3$

Now, compute the limits of the inner integral:

= \cos(x^3 - 1) \left( \frac{9}{2} - \frac{9x^2}{2} \right) = \frac{9}{2} \cos(x^3 - 1) (1 - x^2)$$ ### Step 5: Perform the outer integration Now, integrate with respect to $$x$$: $$\int_0^1 \frac{9}{2} \cos(x^3 - 1) (1 - x^2) \, dx$$ This integral does not have a simple closed form and may require numerical methods to solve. However, we can set it up for numerical evaluation as: $$\frac{9}{2} \int_0^1 \cos(x^3 - 1) (1 - x^2) \, dx$$ At this point, we would typically use a calculator or software to compute the numerical value of this integral. Would you like more details on how to evaluate this numerically, or do you have any specific questions? ### Related questions: 1. How does changing the order of integration affect the limits of a double integral? 2. What is the role of the Jacobian when transforming variables in double integrals? 3. How can numerical methods like Simpson's rule or trapezoidal rule help in evaluating integrals like this one? 4. What are some common strategies to tackle integrals involving trigonometric functions of polynomials? 5. Can this integral be solved using approximation techniques like Taylor series expansion? ### Tip: For integrals with complex functions, numerical integration techniques such as Gaussian quadrature or software like Wolfram Mathematica or MATLAB can be very useful.