Question

Evaluate the integrals. $$ \int_{y=-2}^{1} \int_{x=1}^{2} 8 x y d x d y $$

Short answer

-18

Step-by-step solution

Understand the Problem

The problem asks to evaluate the double integral \[ \int_{y=-2}^{1} \int_{x=1}^{2} 8 x y \, d x \, d y \]. This involves integrating the function \[8xy \] first with respect to \ x\ and then with respect to \ y\ over the given limits.

Integrate with Respect to x

First, evaluate the integral of \[8xy\] with respect to \ x \ while keeping \ y \ constant. \[ \int_{x=1}^{2} 8 x y \, d x \] Evaluate the integral: \[ \int_{x=1}^{2} 8 x y \, d x = 8 y \int_{x=1}^{2} x \, d x = 8 y \left[ \frac{x^2}{2} \right]_{1}^{2} \].

Apply the Limits for x

Next, apply the limits for \ x \ from 1 to 2: \[ 8 y \left( \frac{2^2}{2} - \frac{1^2}{2} \right) = 8 y \left( \frac{4}{2} - \frac{1}{2} \right) = 8 y \left( 2 - 0.5 \right) = 8 y (1.5) = 12 y \].

Integrate with Respect to y

Now, integrate the result \[12 y \] with respect to \ y \ over the limits from -2 to 1: \[ \int_{y=-2}^{1} 12 y \, d y \].

Evaluate the Integral with Respect to y

Evaluate the integral: \[ \int_{y=-2}^{1} 12 y \, d y = 12 \int_{y=-2}^{1} y \, d y = 12 \[ \frac{y^2}{2} \]_{-2}^{1} \].

Apply the Limits for y

Next, apply the limits for \ y \ from -2 to 1: \[ 12 \left( \frac{1^2}{2} - \frac{(-2)^2}{2} \right) = 12 \left( \frac{1}{2} - \frac{4}{2} \right) = 12 \left( 0.5 - 2 \right) = 12 \left( -1.5 \right) =-18 \].

Key concepts

step-by-step integration

Step-by-step integration is a methodical process in calculus used to evaluate integrals, particularly double integrals, by breaking down the problem into manageable steps. Let's walk through this:
First, identify the order of integration. In this exercise, we integrate with respect to \( x \) first, then with respect to \( y \).
Next, handle each integration sequentially:

• Integrate the function with respect to the first variable, keeping the second variable constant.
• Apply the limits for that variable.
• Use the result to integrate with respect to the second variable.
• Apply the limits for the second variable.

Following these steps ensures accurate and systematic evaluation of double integrals.

evaluating integrals

Evaluating integrals refers to the process of finding the value of an integral, which represents the area under a curve for single integrals or the volume under a surface for double integrals. Here’s how to evaluate a double integral:
Identify the function to be integrated and its limits. In our exercise, the function is \( 8xy \), and the limits for \( x \) are from 1 to 2 and for \( y \) from -2 to 1.
Perform the first integral. Integrate \( 8xy \) with respect to \( x \): \[ \int_{x=1}^{2} 8xy \, dx \]. Use the limits for \( x \).
Then, take the result (in this case \( 12y \)), and integrate it with respect to \( y \): \[ \int_{y=-2}^{1} 12y \, dy \].
Apply the limits for \( y \) to obtain the final value.
This process ensures we accurately evaluate the integral and get the correct area or volume.

limits of integration

Limits of integration define the range over which the integration is performed. They are crucial in setting the boundaries within which the function is integrated.
For double integrals, the limits can be expressed as constants or functions. In our given problem:

• The inner limits \( 1 \) to \( 2 \) are for \( x \), defining where to start and stop integrating with respect to \( x \).
• The outer limits \( -2 \) to \( 1 \) are for \( y \), defining the range for integrating the result with respect to \( y \).

Applying the limits:

• First, perform the inner integration within the specified \( x \) limits.
• Plug the upper and lower limits into the resulting formula.
• Next, perform the outer integration within the specified \( y \) limits.
• Finally, substitute the \( y \) limits into the resulting expression.

This structured approach ensures that the integral is evaluated correctly.

multivariable calculus

Multivariable calculus extends calculus to functions of several variables. It includes topics like partial derivatives, multiple integrals, and vector calculus.
For double integrals, typical in multivariable calculus, understanding the function's behavior over a two-dimensional region is key.

The steps to solve include:
• Identifying the function and region of integration.
• Performing multiple integrations step-by-step.
• Defining and applying limits of integration correctly.

Multivariable calculus allows calculating areas, volumes, and other physical quantities over regions in space. Our exercise involves integrating \( 8xy \) over specific limits for \( x \) and \( y \), demonstrating these crucial concepts and techniques. With practice, solving such problems becomes intuitive.