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Chapter 21: Problem 2

$$ \int_{0}^{4} \int_{0}^{y} d x d y $$

Short Answer

Expert verified
The integral evaluates to 8.

Step by step solution

01

Recognize the Order of Integration

The integral is given as \ \ \int_{0}^{4} \int_{0}^{y} dx dy \ which means we first integrate with respect to \(x\) and then \(y\).
02

Integrate the Inner Integral

Integrate \( \int_{0}^{y} dx \) with respect to \(x\). Since the integrand is 1, the integration of 1 with respect to \(x\) is \(x\). Therefore, we need to evaluate: \[ \int_{0}^{y} dx = [x]_{0}^{y} = y - 0 = y \]
03

Integrate the Resulting Function

Now, integrate the result \ (y) with respect to \(y\) over the given limits from 0 to 4: \[ \int_{0}^{4} y dy \]
04

Perform the Outer Integration

Integrate \( \int_{0}^{4} y dy \). The antiderivative of \(y\) with respect to \(y\) is \( \frac{y^2}{2}\). So, we need to evaluate: \[ \int_{0}^{4} y dy = \left[ \frac{y^2}{2} \right]_{0}^{4} = \frac{4^2}{2} - \frac{0^2}{2} = \frac{16}{2} = 8 \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Order of Integration
In double integrals, the order of integration tells us the sequence in which we need to integrate the variables. Here, the integral is \ \int_{0}^{4} \int_{0}^{y} dx dy \ \, so we first integrate with respect to \(x\) and then with respect to \(y\).

The integration order can significantly affect how we approach the problem. If the order were reversed, we would need a different approach. Always pay attention to the order as stipulated in the problem.

When changing the order of integration, keep the integration limits in mind. They must accurately reflect the regions of integration so we can solve correctly.
Inner Integral
The inner integral is the part of the problem that we start with. In this case, \int_{0}^{y} dx\ is the inner integral. This means we integrate with respect to \(x\) while treating \(y\) as a constant.

For our exercise, the inner integral simplifies because we are integrating a constant function (1): \ \int_{0}^{y} dx = x |_{0}^{y} = y - 0 = y

Always verify the limits of the inner integral, as they set the bounds for that specific integration. Determining the inner integral correctly is crucial for the next steps.
Outer Integral
Once we solve the inner integral, we move to the outer integral. In our problem, this is \int_{0}^{4} y dy\, where we integrate the resulting function from the inner integral with respect to \(y\).

For example, after integrating \int_{0}^{y} dx\, we obtained y. We then integrate y with respect to y over the given limits:

\int_{0}^{4} y dy
\ The antiderivative of \( y \) is \( \frac{y^2}{2} \). Evaluate this from 0 to 4:

\left [ \frac{y^2}{2} \right ]_{0}^{4} = \frac{16}{2} - \frac{0}{2} = 8

Properly integrating the outer integral gives us the solution to the problem. It's essential for correctly evaluating the entire double integral.

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