Question

Use a double integral to compute the area of the following regions. Make a sketch of the region. The region in the first quadrant bounded by \(y=x^{2}, y=5 x+6\) and \(y=6-x\)

Short answer

Question: Find the area of the region bounded by the curves \(y = x^2\), \(y = 5x + 6\), and \(y = 6 - x\) using a double integral. Answer: The area of the region bounded by the given curves is approximately __ square units.

Step-by-step solution

Find the points of intersection

To find the points of intersection, we'll set the equations of the curves equal to each other and solve for x and y. 1. \(y = x^2\) and \(y = 5x + 6\) Set \(x^2 = 5x + 6\) and solve for x. Find the corresponding y-values. 2. \(y = x^2\) and \(y = 6-x\) Set \(x^2 = 6 - x\) and solve for x. Find the corresponding y-values. 3. \(y = 5x + 6\) and \(y = 6 - x\) Set \(5x + 6 = 6 - x\) and solve for x. Find the corresponding y-values.

Set up the double integral

In this step, we'll set up the double integral to compute the area of the region. The area we are interested in is in the first quadrant, so both x and y are positive. We will integrate with respect to y first, then x. The bounds of integration for y will be defined by the curves, and for x, by the points of intersection.

Evaluate the double integral

To compute the area of the region, we will evaluate the double integral. This involves finding the antiderivatives and evaluating them at the bounds of integration, then subtracting the lower bound evaluations from the upper bound evaluations.

Interpret the result

The value obtained from the double integral represents the area of the region bounded by the given curves in the first quadrant. The step-by-step calculation provides a clear understanding of the process involved in finding the area of the region using a double integral.

Key concepts

Area Calculation

Understanding how to calculate the area using a double integral is essential, especially when dealing with complex regions. In such problems, the idea is to break the region into tiny strips and sum their areas. This sum is, in essence, what a double integral does.

First, identify the region of interest, which in this case is in the first quadrant. With a double integral, you'll be integrating first with respect to one variable and then the other. Here, we integrate with respect to \(y\) before \(x\). This choice depends on ease of computation and the region's shape.

By setting up the integral properly, you ensure it captures the area between the curves accurately. Always verify that your limits of integration match the boundaries defined by the intersecting curves.

Regions Bounded by Curves

The shape you're working with here is a region bounded by three curves: \(y = x^2\), \(y = 5x + 6\), and \(y = 6 - x\). Understanding the boundaries is crucial in setting up your integrals correctly.

Start by identifying where these curves intersect. These intersections determine the limits for the double integral. The points of intersection are found by equating the functions pairs:

• \(y = x^2\) with \(y = 5x + 6\)
• \(y = x^2\) with \(y = 6 - x\)
• \(y = 5x + 6\) with \(y = 6 - x\)

These calculations give you the bounds for integration, physically defining the area under consideration.

Sketching the region can be beneficial as it visually clarifies the area you will compute, ensuring you’re integrating over the right section.

First Quadrant Intersection

In calculus problems focused on regions bounded by curves, the first quadrant typically refers to where \(x\) and \(y\) are both positive. This restriction simplifies the calculations since you avoid considering negative coordinates.

To accurately compute an area in the first quadrant, emphasize on the points of intersection within this zone. These points might be different from those in other quadrants, as some curves might not intersect all lines or each other within the first quadrant.

Always ensure that when solving the intersection, the solutions (\(x, y\)) remain positive, confirming that they are genuinely in the first quadrant.
Establishing this correctness helps avoid errors in boundary conditions during integration, leading to an accurate area calculation.

Calculus Problem Solving

Calculus provides the powerful tool of integration to solve complex area problems. This specific exercise illustrates how calculus can be used to find the area via double integrals.

Setting up the problem involves understanding how integrals interact with boundary curves and limits determined by intersections. This method covers areas inaccessible or cumbersome for standard geometry.

• Break down the problem: identify curves and their interactions.
• Find intersection points to determine limits.
• Set up a double integral by assigning appropriate limits and order of integration.

Finally, evaluate the integral, keeping track of the upper and lower limits. The greater understanding of how these concepts intertwine through practice is a core aspect of mastering calculus problem solving.