Question

The following integrals can be evaluated only by reversing the order of integration. Sketch the region of integration, reverse the order of integration, and evaluate the integral. $$\int_{0}^{\pi} \int_{x}^{\pi} \sin y^{2} d y d x$$

Short answer

Based on the given integral, we have to evaluate the integral by reversing the order of integration. We have sketched the region of integration, reversed the order, and evaluated the integral. After switching the order of integration, we obtained the new integral as: $$\int_{0}^{\pi} \int_{0}^{y} \sin y^{2} dx dy$$ Using integration by parts and identifying that the remaining integral does not have a simple analytical solution, we concluded that numerical integration methods are needed to find the final result.

Step-by-step solution

Sketch the Region of Integration

First, let's analyze the given limits of integration. We know that $$x$$ ranges from 0 to $$\pi$$ and $$y$$ ranges from $$x$$ to $$\pi$$. This implies that the region we're dealing with is a triangle with vertices (0, 0), ($$\pi$$, 0), and ($$\pi$$, $$\pi$$). Sketch this triangle on a coordinate plane to visualize the region of integration better.

Reverse the Order of Integration

Now, we need to reverse the order of integration. Currently, we have $$dy dx$$, so we want to rewrite the integral in terms of $$dx dy$$. To do this, we'll need to replace the limits of integration for both $$x$$ and $$y$$. We see that $$y$$ ranges from $$x$$ to $$\pi$$. However, in terms of $$x$$, we must think about how the variables vary within the triangle. We see that for a given $$y$$, $$x$$ varies from 0 to $$y$$. Similarly, $$y$$ varies from 0 to $$\pi$$. So, the new limits of integration are: $$\int_{0}^{\pi} \int_{0}^{y} \sin y^{2} dx dy$$

Evaluate the Inner Integral

Now, let's evaluate the inner integral with respect to $$x$$: $$\int \sin y^2 dx = x \sin y^2 + C$$ We can plug in the limits of integration for $$x$$: $$\int_{0}^{y} \sin y^2 dx = (y \sin y^2) - (0 \sin y^2) = y \sin y^2$$

Evaluate the Outer Integral

Now, we can evaluate the outer integral with respect to $$y$$: $$\int_{0}^{\pi} y \sin y^2 dy$$ We can use integration by parts, where $$u = y$$ and $$dv = \sin y^2 dy$$ and follow these steps: 1. $$du = dy$$ 2. $$v = \int \sin y^2 dy$$ (we can't find the exact antiderivative, leave as is) Now, apply integration by parts formula $$\int u dv = uv - \int v du$$: $$\int_{0}^{\pi} y \sin y^2 dy = y\left(\int \sin y^2 dy\right)\Big|_{0}^{\pi} - \int_{0}^{\pi} (\int \sin y^2 dy) dy$$ Unfortunately, the remaining integral does not have a simple analytical solution, and we would need to use numerical integration methods to find the final result.

Key concepts

Integration by Parts

Integration by parts is a powerful technique that helps to solve certain types of integrals by transforming them into a potentially simpler form. It comes in handy particularly when an integral is the product of two functions. The basic formula for integration by parts is given by: \[ \int u \, dv = uv - \int v \, du \]Where:

• \( u \) is a function of the variable of integration.
• \( dv \) is the derivative part that you're integrating.
• \( du \) is the derivative of \( u \).
• \( v \) is the antiderivative of \( dv \).

Applying this method requires a strategic choice of \( u \) and \( dv \), as the wrong choice can lead to more complicated integrals. In our example, we chose \( u = y \), leading us to take \( du = dy \), and \( dv = \sin y^2 dy \). This form helps us reorganize the problem, even though the antiderivative \( \int \sin y^2 dy \) isn't straightforward. Yet, integration by parts allows progress in evaluating complex integrals that other straightforward methods cannot address directly.

Integral Limits

Integral limits refer to the boundaries over which we integrate a function. They define the interval in which the function's value is added up. They are the numbers or expressions that appear at the boundary of the integral sign, often variable dependent. When reversing the order of integration, it becomes crucial to appropriately set new limits.

In the exercise, our limits shifted from \( \int_0^\pi \int_x^\pi \) to \( \int_0^\pi \int_0^y \) because we switched the order to integrate \( dx \) first and then \( dy \). Hence, careful analysis of these integral bounds is essential to accurately solving integrals, particularly in double integrals where regions of integration play a key role. Also, sketching the region enclosed by these limits graphically can aid in a better understanding, ensuring proper setup of the problem and optimized solving process.

Numerical Integration

When dealing with functions that do not possess easy antiderivatives, numerical integration provides a practical approach to obtain approximate values. Numerical integration uses algorithms to compute the value of definite integrals, particularly when analytical solutions are intractable.Common numerical methods include:

• Trapezoidal Rule: approximates the region under the curve as a series of trapezoids.
• Simpson's Rule: uses parabolic arcs instead of straight lines to approximate the curve over intervals.
• Monte Carlo Integration: applies random sampling to estimate the integral, excellent for high-dimensional integrals.

In our case, the integral \( \int \sin y^2 \, dy \) is not straightforward. Therefore, using numerical methods would allow us to estimate the result, offering a practical solution when traditional calculus techniques hit a barrier. These methods also enhance computational power when faced with complex integrations that arise in real-world applications.