To solve the double integral $\int_{0}^{1} \int_{2y}^{2} \cos(x^2) \, dx \, dy$ by changing the order of integration, we'll first need to understand the region of integration and then change the order accordingly.

Step 1: Describe the Region of Integration

The integral is given as:

$\int_{0}^{1} \int_{2y}^{2} \cos(x^2) \, dx \, dy$

The inner integral limits $2y$ to 2 indicate the bounds for $x$, while the outer integral limits 0 to 1 indicate the bounds for $y$.

For $y$ from 0 to 1:

• $x$ ranges from $2y$ to 2.

This describes a region in the $xy$-plane where:

1. $0 \leq y \leq 1$,
2. $2y \leq x \leq 2$.

Step 2: Expressing $y$ in Terms of $x$

Next, let's express $y$ in terms of $x$:

• From $x = 2y$, we get $y = \frac{x}{2}$.
• $y$ ranges from 0 to 1, so $x$ ranges from 0 to 2.

For fixed $x$ between 0 and 2:

• $y$ ranges from $0$ to $\frac{x}{2}$.

Step 3: Write the Integral with the New Order

Changing the order of integration, the integral becomes:

$\int_{0}^{2} \int_{0}^{\frac{x}{2}} \cos(x^2) \, dy \, dx$

Step 4: Simplify and Evaluate

Notice that $\cos(x^2)$ does not depend on $y$, so the inner integral is straightforward:

$\int_{0}^{2} \left[ \cos(x^2) \cdot y \right]_{y=0}^{y=\frac{x}{2}} \, dx$

This simplifies to:

$\int_{0}^{2} \frac{x}{2} \cos(x^2) \, dx$

Finally, the integral is:

$\frac{1}{2} \int_{0}^{2} x \cos(x^2) \, dx$

Step 5: Perform the Substitution

To solve this integral, use the substitution $u = x^2$, hence $du = 2x \, dx$.

$\frac{1}{2} \int_{0}^{2} x \cos(x^2) \, dx = \frac{1}{4} \int_{0}^{4} \cos(u) \, du$

Evaluating this:

$\frac{1}{4} \left[ \sin(u) \right]_{0}^{4} = \frac{1}{4} (\sin(4) - \sin(0)) = \frac{\sin(4)}{4}$

So, the final result is:

$\frac{\sin(4)}{4}$

Final Answer:

$\int_{0}^{1} \int_{2y}^{2} \cos(x^2) \, dx \, dy = \frac{\sin(4)}{4}$

Would you like more details or have any questions?

Related Questions:

1. How do you perform a change of variables in double integrals?
2. What are the common methods for evaluating double integrals?
3. How does the region of integration affect the order of integration?
4. Can you explain the geometric interpretation of changing the order of integration?
5. What are some common trigonometric integrals and their solutions?

Tip: When changing the order of integration, always sketch the region to clearly understand the limits.