Problem 40 - Advanced Engineering Mathematics

Question

Evaluate the given iterated integral by reversing the order of integration. $$ \int_{0}^{4} \int_{\sqrt{y}}^{2} \sqrt{x^{3}+1} d x d y $$

Step-by-Step Solution

Answer

To change the order of integration, use limits: \( \int_{0}^{2} \int_{x^2}^{4} \sqrt{x^3+1} \, dy \, dx \).

Step 1: Understand the Current Limits of Integration

The given integral is \( \int_{0}^{4} \int_{\sqrt{y}}^{2} \sqrt{x^{3}+1} \, d x \, d y \). The outer integral with respect to \( y \) ranges from 0 to 4, while the inner integral with respect to \( x \) ranges from \( \sqrt{y} \) to 2. This describes a region in the xy-plane.

Step 2: Sketch the Region of Integration

To reverse the order of integration, we first need to understand the region we're integrating over. Draw the limits given:- \( x = \sqrt{y} \) describes a parabola.- \( x = 2 \) is a vertical line segment.- The limits of \( y \) from 0 to 4 form a bounded region to the right of the parabola and below the line \( x = 2 \).

Step 3: Determine the New Integration Limits

Now we change the order of integration. In the region described:- For each fixed value of \( x \), \( y \) will vary from \( y = x^2 \) (the parabola) to \( y = 4 \) (the upper limit from the given integral).- \( x \) varies from 0 to 2 in this range.

Step 4: Set Up the New Iterated Integral

Using the new limits:- The outer integral will be with respect to \( x \) from 0 to 2.- The inner integral will be with respect to \( y \) from \( x^2 \) to 4.The iterated integral now becomes:\[\int_{0}^{2} \int_{x^2}^{4} \sqrt{x^{3}+1} \, d y \, d x\]

Step 5: Evaluate the Inner Integral

Consider the inner integral \( \int_{x^2}^{4} \sqrt{x^{3} + 1} \, dy \). Since \( \sqrt{x^{3} + 1} \) is treated as a constant with respect to \( y \), the integral is simply the product of \( \sqrt{x^{3} + 1} \) and the difference \( 4 - x^2 \):\[\sqrt{x^{3} + 1} \cdot (4 - x^2)\]

Step 6: Evaluate the Outer Integral

Now integrate with respect to \( x \) from 0 to 2:\[\int_{0}^{2} \sqrt{x^{3} + 1} (4 - x^2) \, d x\]This requires applying standard integration techniques, such as polynomial expansion and, potentially, substitution methods or numerical calculations if necessary.

Key Concepts

Order of Integration

The order of integration in an iterated integral tells us the sequence in which the variables should be integrated. Reversing this order can simplify the computation or make it possible to integrate certain functions. For example, if you start with \(\int_{0}^{4} \int_{\sqrt{y}}^{2} \sqrt{x^{3}+1} \, dx \, dy\), the original order of integration is with respect to \(x\) (inside) and \(y\) (outside). Changing this order involves integrating with respect to \(y\) first and then \(x\). The reversed integral is \[\int_{0}^{2} \int_{x^2}^{4} \sqrt{x^{3}+1} \, dy \, dx\]Switching the order allows us to redefine the region of integration and often results in a more manageable integration task.

Limits of Integration

The limits of integration specify the boundaries for each variable in the iterated integral. In our original problem, these limits are \(x = \sqrt{y}\) to \(x = 2\) and \(y = 0\) to \(y = 4\). To reverse the order of integration, it's necessary to redefine these limits according to the new order.

• For the new integral, \(y\) changes from \(x^2\) to \(4\) for every fixed \(x\).
• The values for \(x\) now range from \(0\) to \(2\).

Adjusting these limits accurately is crucial, as they define the region over which the function is integrated. Mistakes can lead to incorrect evaluations, so careful analysis of the region is essential.

Integration Techniques

To evaluate iterated integrals, especially after reversing the order of integration, various integration techniques are applied. These techniques include:

• Polynomial expansion: used when integrands are polynomial expressions that require expanding for easier integration.
• Substitution: a strategic change of variables may make a complex integral more straightforward to solve.
• Numerical calculations: when analytical solutions are difficult, numerical methods provide approximate solutions.

In our specific integral, after reversing the order, the inner integral was integrated as a constant with respect to \(y\), simplifying the integration.Evaluating the transformed iterated integral may also necessitate combinations of these techniques to obtain the final result.

Region of Integration

The region of integration is the area in the coordinate plane over which we are integrating the function. This region can be described using the limits of the iterated integral. In the given exercise:

• The initial area analyzed was bounded by \(y = x^2\) and \(x = 2\), along with horizontal bounds from \(y = 0\) to \(y = 4\).
• Reversing the order redefines this region in terms of fixed values of \(x\), where \(y\) varies from \(x^2\) to \(4\).

Drawing or sketching the region helps in visually confirming the limits make sense and in understanding how they relate to one another. This makes it easier to set new boundaries effectively when changing the integration order, ensuring accurate evaluations.