Problem 25 - Applied Calculus: For Business, Economics, and the Social and Life Sciences

Question

Evaluate the given double integral for the specified region \(R\).\(\iint_{R} 3 x y^{2} d A\), where \(R\) is the rectangle bounded by the lines \(x=-1, x=2, y=-1\), and \(y=0\).

Step-by-Step Solution

Answer

The value of the double integral is \( \frac{3}{2}\).

Step 1: Understand the Limits of Integration

Identify the region of integration. Here, the region \(R\) is bounded by \(x = -1\) to \(x = 2\) and \(y = -1\) to \(y = 0\).

Step 2: Set up the Double Integral

Write the double integral with the appropriate limits of integration: \(\iint_{R} 3xy^{2} \, dA = \int_{-1}^{2} \int_{-1}^{0} 3xy^{2} \, dy \, dx\) or vice versa.

Step 3: Integrate with Respect to y

Compute the inner integral with respect to \(y\): \(\int_{-1}^{0} 3xy^{2} \, dy = 3x \int_{-1}^{0} y^{2} \, dy\). Use the antiderivative of \(y^{2}\) to get \(3x \left[ \frac{y^{3}}{3} \right]_{-1}^{0} = 3x \left(0 - (- \frac{1}{3})\right) = x\).

Step 4: Integrate with Respect to x

Now, integrate the result with respect to \(x\): \(\int_{-1}^{2} x \, dx\). Use the antiderivative of \(x\) to get \( \left[ \frac{x^{2}}{2} \right]_{-1}^{2} = \left( \frac{2^{2}}{2} \right) - \left( \frac{(-1)^{2}}{2} \right) = 2 - \frac{1}{2} = \frac{3}{2}\).

Step 5: Combine the Results

The value of the given double integral is \( \frac{3}{2} \).

Key Concepts

Integration Limits

When solving a double integral, understanding the integration limits is key. These limits define where the integration starts and ends along each axis. For the double integral \(\iint_{R} 3 x y^{2} \, dA\), the region \(R\) is bounded by \(x = -1\) to \(x = 2\) and \(y = -1\) to \(y = 0\). It's essential to correctly identify these bounds to set up the integral properly.
By correctly analyzing the region on an \(xy\)-plane, you make sure that your integral covers the entire specified region. Begin by writing down the inequalities for \(x\) and \(y\):

• \(-1 \leq x \leq 2\)
• \(-1 \leq y \leq 0\)

These inequalities will serve as the limits for our iterated integrals.

Region of Integration

The region of integration is the area over which you are integrating. It is essential to clearly visualize this region to correctly set up the double integral.
In our case, the region \(R\) is a rectangle in the \(xy\)-plane. It's bounded by lines parallel to the axes:

• The vertical lines \(x = -1\) and \(x = 2\)
• The horizontal lines \(y = -1\) and \(y = 0\)

This rectangular region is simple to describe and handle mathematically, making it straightforward to identify the integration limits for both \(x\) and \(y\). Understanding this setup helps us to break down the double integral into more manageable parts.

Iterated Integrals

Double integrals can be evaluated using iterated integrals. This means we first integrate with respect to one variable, and then use that result to integrate with respect to the second variable.
For our exercise, the iterated integral setup looks like this:

• \( \iint_{R} 3xy^{2} \, dA = \int_{-1}^{2} \int_{-1}^{0} 3xy^{2} \, dy \, dx \)

Here, we integrate the function \(3xy^{2}\) first with respect to \(y\), while treating \(x\) as constant. After computing this inner integral, we then integrate the resulting expression with respect to \(x\).
This step-by-step breakdown simplifies the complex process of computing a double integral by focusing on one variable at a time.

Antiderivatives

Antiderivatives play a crucial role when solving integrals. An antiderivative of a function is another function whose derivative is the original. For our integral, we need to find antiderivatives twice.
First, for the inner integral with respect to \(y\):\[ \int_{-1}^{0} 3xy^{2} \, dy = 3x \left[ \frac{y^{3}}{3} \right]_{-1}^{0} = 3x \left( 0 - \left( - \frac{1}{3} \right) \right) = x \]
Here, \( \frac{y^{3}}{3} \) is the antiderivative of \( y^{2} \).
Next, for the outer integral with respect to \(x\):\[ \int_{-1}^{2} x \, dx = \left[ \frac{x^{2}}{2} \right]_{-1}^{2} = \left( \frac{2^{2}}{2} \right) - \left( \frac{(-1)^{2}}{2} \right) = 2 - \frac{1}{2} = \frac{3}{2} \]
The antiderivative of \( x \) is \( \frac{x^{2}}{2} \).
Using antiderivatives simplifies the process of integrating complex functions by breaking them down into more manageable parts.