Question

Write an iterated integral of a continuous function \(f\) over the following regions. The region bounded by \(y=2 x+3, y=3 x-7,\) and \(y=0\)

Short answer

Based on the step by step solution provided, write a short answer question. Question: Find the iterated integral of a continuous function 𝑓 over the region bounded by 𝑦=2𝑥+3, 𝑦=3𝑥−7, and 𝑦=0 in the dydx order of integration. Answer: $$\int_{-\frac{3}{2}}^{\frac{7}{3}} \int_{0}^{2x + 3} f(x,y) \: dy\: dx$$

Step-by-step solution

Find the Intersection Points

We first find the points where the lines intersect so that we can determine the bounds of integration. - Intersection of \(y=0\) and \(y=2x + 3\): \(0 = 2x + 3\) \(x = -\frac{3}{2}\) - Intersection of \(y=0\) and \(y=3x - 7\): \(0 = 3x - 7\) \(x = \frac{7}{3}\) - Intersection of \(y=2x + 3\) and \(y=3x - 7\): \(2x + 3 = 3x -7\) \(x = 10\) \(y = 2(10) + 3 = 23\) So the intersection points are \((-\frac{3}{2}, 0)\), \((\frac{7}{3}, 0)\), and \((10, 23)\).

Determine the Order of Integration

Since the given functions describe lines, we can choose either order of integration (dxdy or dydx) without difficulty. In this case, we will use dydx.

Write the Iterated Integral

We will write the iterated integral of the continuous function \(f\) over the given region, using the dydx order of integration and the intersection points found in Step 1. - The \(x\) bounds of integration are from \(x = -\frac{3}{2}\) to \(x = \frac{7}{3}\). - For a given \(x\) value between \(-\frac{3}{2}\) and \(\frac{7}{3}\), - The lower bound of \(y\) is \(y=0\), - The upper bound of \(y\) is the line connecting the points \((-\frac{3}{2}, 0)\) and \((10, 23)\), which can be found using the slope and a point. The slope is \(\frac{23}{\frac{23}{2}} = 2\). So, the equation of the line is \(y = 2x + 3\). The iterated integral is: $$\int_{-\frac{3}{2}}^{\frac{7}{3}} \int_{0}^{2x + 3} f(x,y) \: dy\: dx$$

Key concepts

Continuous Function

A continuous function is like a smooth road with no bumps or breaks. In mathematics, a function is continuous when there are no sudden jumps or drops in its graph. For example, if you are drawing the curve of the function and you can do it without lifting your pen off the paper, then it's continuous. Continuous functions are essential in calculus because they make it possible to apply integration and differentiation reliably.

In the context of iterated integrals, having a continuous function ensures that the calculations we perform over a region give meaningful and accurate results. This is because continuous functions ensure that every value between the bounds is accounted for smoothly. This stability is crucial when performing operations like finding the area or volume under the curve, or in this case, within the region defined by the bounds of integration.

Bounds of Integration

When we talk about bounds of integration, we refer to the values that define the limits of the region over which we are integrating. In simple terms, these are the "starting" and "ending" points of your journey along the x-axis and y-axis.

In our problem, the bounds for the iterated integral are found by understanding where the lines intersect. For the outer integral (corresponding to the x-direction), the bounds are from \(-\frac{3}{2}\) to \(\frac{7}{3}\). For the inner integral, which corresponds to the y-direction for each specific x value, the bounds are determined by the lines \(y=0\) and \(y = 2x + 3\).

• The x-bounds describe the horizontal span of the region.
• The y-bounds describe the vertical limits from step 3's integration setup.

Understanding these bounds helps us know exactly what part of the function we are analyzing, ensuring that we cover the area defined by these intersecting lines fully.

Order of Integration

The order of integration refers to the sequence in which we perform integration when working with multiple variables. In iterated integrals, this means deciding whether to integrate with respect to x first and then y, or vice versa.

For this problem, we have chosen the dydx order. This means integrating first with respect to y, and then with respect to x. When the conditions of the region or function allow it, like in this exercise, the order can be chosen based on convenience or simplicity.

• Choose dydx if integrating with respect to y first simplifies the process.
• Conversely, choose dxdy if integrating with respect to x first is simpler.

Changing the order of integration can sometimes make the calculation more straightforward, especially if the region is more intuitively divided along one axis. It’s important to note that the actual numerical answer should remain the same regardless of the order chosen.

Intersection Points

Finding intersection points is like solving a puzzle to understand where different geometric shapes meet. These are usually the points where equations that describe lines or curves cross each other. In many calculus problems, these intersection points form the boundaries for integration.

In this exercise, the intersection points have been identified by solving for where the lines \(y=2x+3\), \(y=3x-7\), and \(y=0\) intersect.

• Solving \(y=2x+3\) and \(y=0\) gives the point \((-\frac{3}{2},0)\).
• Solving \(y=3x-7\) and \(y=0\) yields \((\frac{7}{3},0)\).
• Finally, solving \(y=2x + 3\) and \(y=3x - 7\) results in \((10, 23)\).

These intersection points are critical because they help define the region over which we apply the integrals. By identifying these points accurately, we can set precise bounds for integration, ensuring that the calculation considers the right portion of the function.