Evaluate Double Integral: Step-by-Step Guide

Guys, let's dive into this fascinating problem of evaluating a double integral. We're given the double integral $\iint_D x^2 dxdy$, where D is a region bounded by the curves $xy = 16$, $y = x$, $y = 0$, and $x = 8$. Sounds like a geometric puzzle, right? Well, that's essentially what it is! To solve this, we need to first understand the region D and then set up the integral limits correctly. Let's break it down step-by-step.

Understanding the Region of Integration

First off, let's visualize the region D. We have four boundary curves:

1. xy = 16: This is a hyperbola. If you sketch it out, you'll see it lies in the first and third quadrants, but we're mainly concerned with the first quadrant since the other boundaries restrict us there.
2. y = x: A straight line passing through the origin with a slope of 1. Super straightforward.
3. y = 0: This is the x-axis. Our region is bounded below by this.
4. x = 8: A vertical line at x = 8. This gives us a clear right boundary.

Now, let's find the intersection points of these curves. These points will help us define the limits of integration. The intersection of y = x and xy = 16 occurs when $x * x = 16$, which gives us $x^2 = 16$. Thus, x can be ±4. Since we are in the first quadrant (bounded by x = 8 and y = 0), we take x = 4, and thus y = 4. So, one intersection point is (4, 4).

The intersection of xy = 16 and x = 8 occurs when $8y = 16$, which gives us y = 2. So, another intersection point is (8, 2).

The intersection of y = x and x = 8 occurs at the point (8, 8).

Looking at these intersections and the curves, we realize that the region D is divided into two sub-regions. This is because the lower boundary changes from the line y = x to the hyperbola xy = 16 within the interval. Let's call these sub-regions D₁ and D₂.

• D₁: Bounded by y = x, y = 0, and x = 8 up to the intersection point (4, 4).
• D₂: Bounded by xy = 16, y = 0, and x = 8 from the intersection point (4, 4) to (8, 2).

Setting Up the Double Integral

Since our region D is best described by splitting it into D₁ and D₂, we'll express the double integral as a sum of two double integrals:

$$\iint_D x^2 dxdy = \iint_{D_1} x^2 dxdy + \iint_{D_2} x^2 dxdy$$

Let's set up the integrals for D₁ and D₂ individually.

Integral over D₁

In D₁, y varies from 0 to x, and x varies from 0 to 4. Thus, the double integral over D₁ is:

$$\iint_{D_1} x^2 dxdy = \int_{0}^{4} \int_{0}^{x} x^2 dy dx$$

Integral over D₂

In D₂, y varies from 0 to 16/x, and x varies from 4 to 8. Thus, the double integral over D₂ is:

$$\iint_{D_2} x^2 dxdy = \int_{4}^{8} \int_{0}^{\frac{16}{x}} x^2 dy dx$$

Now, let's calculate these integrals.

Evaluating the Integrals

Evaluating the Integral over D₁

First, we evaluate the inner integral with respect to y:

$$\int_{0}^{x} x^2 dy = x^2 \int_{0}^{x} dy = x^2[y]_{0}^{x} = x^2(x - 0) = x^3$$

Now, we evaluate the outer integral with respect to x:

$$\int_{0}^{4} x^3 dx = \left[\frac{x^4}{4}\right]_{0}^{4} = \frac{4^4}{4} - \frac{0^4}{4} = \frac{256}{4} = 64$$

So, the integral over D₁ is 64.

Evaluating the Integral over D₂

First, we evaluate the inner integral with respect to y:

$$\int_{0}^{\frac{16}{x}} x^2 dy = x^2 \int_{0}^{\frac{16}{x}} dy = x^2[y]_{0}^{\frac{16}{x}} = x^2\left(\frac{16}{x} - 0\right) = 16x$$

Now, we evaluate the outer integral with respect to x:

$$\int_{4}^{8} 16x dx = 16\int_{4}^{8} x dx = 16\left[\frac{x^2}{2}\right]_{4}^{8} = 16\left(\frac{8^2}{2} - \frac{4^2}{2}\right) = 16\left(\frac{64}{2} - \frac{16}{2}\right) = 16(32 - 8) = 16(24) = 384$$

So, the integral over D₂ is 384.

Final Result

Adding the results from D₁ and D₂, we get the final result:

$$\iint_D x^2 dxdy = \iint_{D_1} x^2 dxdy + \iint_{D_2} x^2 dxdy = 64 + 384 = 448$$

Therefore, the value of the double integral $\iint_D x^2 dxdy$ over the region D is 448.

Double integrals are a cornerstone of multivariable calculus, allowing us to compute volumes, areas, and other quantities over two-dimensional regions. But, tackling these integrals can seem daunting at first. Fear not! Let’s break down the process into manageable steps, making it crystal clear how to evaluate these integrals. This guide will focus on the practical steps and insights needed to confidently solve double integrals, ensuring you're well-equipped to handle any problem thrown your way.

Step 1: Understand the Region of Integration

The first, and arguably the most crucial, step in evaluating a double integral is understanding the region over which you are integrating. The region, often denoted as R or D, defines the limits of your integration. Visualizing this region is paramount. If you can’t picture it, you’ll struggle to set up the integral correctly.

Sketch the Region

Always start by sketching the region. The boundaries are usually given by equations of curves or lines. For instance, you might have a region bounded by $y = x^2$, $y = 4$, $x = 0$, and $x = 2$. Sketching these on a coordinate plane helps you see the shape of the region and where these curves intersect.

Identify the Boundaries

Next, identify which curves form the boundaries of your region. Pay close attention to intersection points; these will often serve as your limits of integration. For example, the intersection points of $y = x^2$ and $y = 4$ are critical for defining the range of your x and y values.

Determine the Order of Integration

Deciding the order of integration (dx dy or dy dx) is crucial. This choice can significantly affect the complexity of the integral. Generally:

• If the region is easily described with y varying between two functions of x (i.e., $g_1(x) \leq y \leq g_2(x)$) and x varying between two constants (i.e., $a \leq x \leq b$), then integrating dy dx is a good choice.
• Conversely, if the region is easily described with x varying between two functions of y (i.e., $h_1(y) \leq x \leq h_2(y)$) and y varying between two constants (i.e., $c \leq y \leq d$), then integrating dx dy is preferable.

Sometimes, one order is significantly easier than the other due to the nature of the functions involved. If one order leads to integrals that are hard to solve, switch the order! Trust me, guys, it's worth the effort to explore both options.

Step 2: Set Up the Double Integral

Once you understand the region and have chosen the order of integration, it’s time to set up the integral. This involves defining the limits of integration based on the boundaries of your region.

Determine the Limits of Integration

• For dy dx, the inner integral (with respect to y) will have limits that are functions of x (i.e., $g_1(x)$ and $g_2(x)$), and the outer integral (with respect to x) will have constant limits (i.e., a and b).
• For dx dy, the inner integral (with respect to x) will have limits that are functions of y (i.e., $h_1(y)$ and $h_2(y)$), and the outer integral (with respect to y) will have constant limits (i.e., c and d).

So, a double integral might look like:

$$\iint_R f(x, y) dA = \int_{a}^{b} \int_{g_1(x)}^{g_2(x)} f(x, y) dy dx$$

or

$$\iint_R f(x, y) dA = \int_{c}^{d} \int_{h_1(y)}^{h_2(y)} f(x, y) dx dy$$

where f(x, y) is the function you are integrating.

Write Out the Integral

Write out the integral explicitly with the correct limits and order. This step is critical for avoiding mistakes later. Double-check that your limits match your region and the order of integration you’ve chosen. It's like setting the foundation for a building; if it's shaky, the whole structure will suffer!

Step 3: Evaluate the Inner Integral

Now comes the fun part: evaluating the integral. Start with the inner integral. Treat the other variable as a constant during this step.

Integrate with Respect to the Inner Variable

If you are integrating dy dx, integrate f(x, y) with respect to y, treating x as a constant. If you are integrating dx dy, integrate f(x, y) with respect to x, treating y as a constant. This will result in a function of the outer variable.

For example, if you have:

$$\int_{a}^{b} \int_{g_1(x)}^{g_2(x)} f(x, y) dy dx$$

First, compute:

$$\int_{g_1(x)}^{g_2(x)} f(x, y) dy = F(x, y) \Big|_{y=g_1(x)}^{y=g_2(x)} = F(x, g_2(x)) - F(x, g_1(x))$$

where F(x, y) is the antiderivative of f(x, y) with respect to y. The result is a function of x only.

Substitute the Limits

After finding the antiderivative, substitute the limits of integration. The result should be a function of the outer variable (or a constant, if you're lucky!).

Step 4: Evaluate the Outer Integral

Finally, evaluate the outer integral. This is a single integral, so you should be more familiar with these.

Integrate with Respect to the Outer Variable

Integrate the result from the inner integral with respect to the outer variable. This will give you a numerical value, which is the result of the double integral.

For example, continuing from the previous step:

$$\int_{a}^{b} [F(x, g_2(x)) - F(x, g_1(x))] dx = G(x) \Big|_{x=a}^{x=b} = G(b) - G(a)$$

where G(x) is the antiderivative of $F(x, g_2(x)) - F(x, g_1(x))$ with respect to x. The final result is a number.

Substitute the Limits and Simplify

Substitute the limits of integration and simplify to get your final answer. This is where careful arithmetic comes into play. Double-check your calculations to avoid silly mistakes!

Even with a solid understanding of the process, it’s easy to stumble. Here are some common pitfalls and how to dodge them.

1. Incorrect Limits of Integration: This is a biggie. Always, always, sketch the region and make sure your limits correspond to the boundaries. If you switch the order of integration, remember to rewrite the limits accordingly.
2. Forgetting to Treat Variables as Constants: When integrating the inner integral, treat the outer variable as a constant. Don't mix them up! This is a classic mistake that can lead to a totally wrong answer.
3. Algebraic Errors: Simple arithmetic or algebraic mistakes can derail the entire process. Take your time, double-check each step, and use a calculator if necessary. Seriously, it's worth it.
4. Choosing the Wrong Order of Integration: Sometimes, one order of integration leads to a much simpler integral. If you’re struggling with one order, try the other. It might just save your day!

To truly master double integrals, you need practice and a few strategic tips up your sleeve. Here are some to help you along the way:

• Practice Regularly: The more you practice, the more comfortable you'll become with the process. Start with simpler integrals and gradually work your way up to more complex problems.
• Use Visual Aids: Sketching the region of integration is crucial. If you're struggling to visualize the region, use graphing software or online tools to help. Desmos, GeoGebra, and Wolfram Alpha can be your best friends here, guys.
• Check Your Answers: If possible, use software or online calculators to check your answers. This can help you identify mistakes and reinforce correct techniques.
• Break Down Complex Problems: If a problem seems overwhelming, break it down into smaller, more manageable steps. Focus on understanding each step before moving on to the next.
• Understand the Geometry: Double integrals are often used to find areas and volumes. Connecting the mathematical process with the geometric interpretation can deepen your understanding and make problem-solving more intuitive.

By following these steps, avoiding common mistakes, and practicing regularly, you'll be well on your way to mastering double integrals. Remember, it's a journey, not a sprint. Keep practicing, and you'll get there! So, go forth and conquer those integrals!

Alright, guys, we've successfully evaluated the double integral $\iint_D x^2 dxdy$, finding it to be 448. This involved understanding the region of integration, splitting it into sub-regions, setting up the integrals correctly, and carefully evaluating each integral. Double integrals can seem complex, but with a systematic approach and a clear understanding of the region, you can tackle them with confidence. So keep practicing, and you'll become a pro in no time!