Question

Evaluate the following iterated integrals. $$\int_{0}^{2} \int_{0}^{1} x^{5} y^{2} e^{x^{3} y^{3}} d y d x$$

Short answer

Question: Evaluate the iterated integral $$\int_{0}^{2} \int_{0}^{1} x^{5} y^{2} e^{x^{3} y^{3}} d y d x$$ Answer: The integral evaluates to $$\frac{1}{3} \int_{0}^{2} \frac{e^{x^3}-1}{x} dx$$

Step-by-step solution

Evaluate the inner integral with respect to y

The inner integral is as follows: $$\int_{0}^{1} x^{5} y^{2} e^{x^{3} y^{3}} d y$$ To solve it, we will use a substitution method. Let's substitute \(u = x^3 y^3\), then \(du = 3x^3y^2 dy\). So, \(dy = \frac{1}{3x^3y^2}du\). Also, when \(y=0\), \(u=0\), and when \(y=1\), \(u=x^3\). Now, substitute these values into the inner integral: $$\int_{0}^{x^3} x^{2} e^{u} \frac{1}{3x^3} du$$ After simplifying: $$\frac{x^2}{3x^3} \int_{0}^{x^3} e^u du$$ Now, we can easily evaluate this integral as follows: $$\frac{x^2}{3x^3}\left[ e^u \right]_{0}^{x^3} $$ Which simplifies to: $$\frac{x^2}{3x^3}(e^{x^3}-1)$$

Evaluate the outer integral with respect to x

Now, we need to evaluate the outer integral using the expression we found in step 1: $$\int_{0}^{2} \frac{x^2}{3x^3}(e^{x^3}-1) d x$$ Simplifying the expression and factoring out the constant: $$\frac{1}{3} \int_{0}^{2} \frac{e^{x^3}-1}{x} dx$$ Unfortunately, this integral does not have a simple closed-form solution in terms of elementary functions. Therefore, we will leave our answer in terms of the unsolved integral: $$\frac{1}{3} \int_{0}^{2} \frac{e^{x^3}-1}{x} dx$$

Key concepts

Substitution Method in Integrals

The substitution method is a crucial technique in integration, especially helpful when dealing with complex functions. In this context, it simplifies the integration process by changing the variables.
By substituting one set of variables for another, you can transform the integral into a simpler form. This is particularly beneficial when the original integral involves difficult to manage expressions.
In our example, the substitution method involves setting a new variable, say, \(u\), to be \(x^3 y^3\). Consequently, the differential \(dy\) changes accordingly to reflect this new variable in terms of \(du\).

• Choose a substitution \(u\) which simplifies the integration.
• Find \(du\) by differentiating \(u\) with respect to \(y\).
• Convert the limits of integration to the new variable \(u\).
• Replace the original integral in \(y\) with its equivalent in \(u\) and integrate.

By applying this approach, the tricky parts of an integrand can be transformed, making the integral more accessible.

Multiple Integration

Multiple integration deals with integrating functions of several variables, typically involving more than one integral, known as iterated integrals. This is frequently seen in problems evaluating areas and volumes.
Functions of multiple variables describe surfaces and volumes, and integrating them provides useful information about their extent in space. In an iterated integral like the given one, integration proceeds along one variable first, followed by the next.
The process involves:

• Determining the order of integration, which can affect the complexity of the calculations but not the final result.
• Evaluating the inner integral first, holding the other variables constant, and using any necessary integration techniques, such as substitution or integration by parts.
• Using the result from the inner integration in the outer integral.

This layered approach can effectively tackle otherwise daunting multivariable functions. Mastering multiple integration is invaluable in fields like physics and engineering.

Change of Variables in Integration

The change of variables technique is a natural extension of substitution methods, especially in higher dimensions. It streamlines integral evaluation by switching to a coordinate system in which the function is easier to integrate.
This technique is akin to reshaping the problem's landscape to one that is easier to work with. When integrals are easier to compute in one coordinate system over another, a change of variables is warranted.
Key steps include:

• Selecting a new set of variables such that the integral expression becomes simpler. This could involve polar coordinates for circular symmetries or logarithmic transformations.
• Computing the Jacobian determinant, which accounts for the "stretching" effect that the transformation imparts on area elements.
• Ensuring that the limits of integration are correctly transformed to reflect the new variables.

Through this approach, seemingly complex integrals can be brought into a more manageable form, showcasing a powerful application of mathematical creativity to solve real-world problems.