Question

Use a double integral to compute the area of the following regions. Make a sketch of the region. The region bounded by the parabola \(y=x^{2}\) and the line \(y=4\)

Short answer

Answer: The area of the region bounded by the parabola \(y=x^2\) and the line \(y=4\) is 16 square units.

Step-by-step solution

Find the intersection points

To find the intersection points of the parabola \(y=x^2\) and the line \(y=4\), set the two functions equal to each other and solve for \(x\): \(x^2 = 4 \Rightarrow x = \pm 2\). So, the intersection points are \((-2, 4)\) and \((2, 4)\).

Set up the double integral for the area of the region

Since we are dealing with Cartesian coordinates, we will integrate with respect to \(x\) first and then with respect to \(y\). The limits of integration for the x-variable will be from \(-2\) to \(2\), and that for the y-variable will be from \(x^2\) to \(4\). The area of the region can be found using the double integral as follows: \(A = \int\int_{R} dA = \int_{-2}^{2}\int_{x^2}^{4} dy\,dx\), where \(R\) is the region bounded by the parabola and the line.

Evaluate the double integral

Now, we will evaluate the double integral to compute the area: \(A = \int_{-2}^{2}\int_{x^2}^{4} dy\,dx = \int_{-2}^{2} (4 - x^2) dx\) To evaluate the integral, we will first integrate with respect to \(x\): \(A = \left[ 4x - \dfrac{x^3}{3} \right]_{-2}^{2}\) Now, plug in the limits of integration and subtract: \(A = \left[ (4(2) - \dfrac{2^3}{3}) - (4(-2) - \dfrac{(-2)^3}{3}) \right]\) \(A = \left[ (8 - \dfrac{8}{3}) - (-8 + \dfrac{8}{3}) \right]\) \(A = 16 - \dfrac{16}{3} = \dfrac{48}{3}=16\)

Sketch the region

Sketch the parabola \(y = x^2\) and the line \(y = 4\) on the same graph. Mark the intersection points \((-2,4)\) and \((2,4)\). Shade the region bounded by the parabola and the line. This is the region whose area we calculated in the previous steps. In conclusion, the area of the region bounded by the parabola \(y=x^2\) and the line \(y=4\) is 16 square units.

Key concepts

Area Calculation

Calculating the area of a region using a double integral is an essential technique in calculus. When you want to find the area between a curve, like a parabola, and a straight line, you use integration. Specifically, a double integral helps you account for the entire area in two-dimensional space.
In the given problem, the region whose area is to be calculated is between the parabola defined by the equation \(y = x^2\) and the line \(y = 4\). To find this area:

• Determine the boundaries of the region.
• Set up an integral with these boundaries.
• Evaluate the integral to find the total area.

For the functions given, the boundaries in the x-direction are from \(-2\) to \(2\), with limits in the y-direction determined by the curves: from \(x^2\) to \(4\). The area is calculated by integrating these limits, producing an area of 16 square units.

Parabola

A parabola is a symmetrical, U-shaped curve that can open upwards or downwards in the Cartesian plane. In this exercise, the parabola is represented by the equation \(y = x^2\). This specific equation indicates an upward-opening parabola.
Characteristics of this parabola include:

• The vertex, or the lowest point on the parabola, is at the origin (0,0).
• The parabola is symmetrical along the y-axis; for any positive x-value, it has a corresponding negative x-value with the same y-value.
• The further x moves away from the origin, the larger y becomes, indicating an increase in the width of the parabola.

Understanding the shape and properties of the parabola helps visualize where it intersects with other lines or curves, in this case, the line \(y = 4\).

Bounded Region

A bounded region in mathematics refers to an enclosed area defined by curves and lines on a graph. In this exercise, the region is bounded by the parabola \(y = x^2\) and the line \(y = 4\).
To define a bounded region:

• Identify where the curves intersect to establish the region's extreme points.
• Analyze each boundary and ascertain the limits these boundaries impose in both the x and y directions.
• Recognize that this specific region looks like a "cap" segment of the parabola sliced off by the line at \(y = 4\).

Once these boundaries and intersections are clearly identified, setting up integrals to find the area becomes a straightforward task.

Cartesian Coordinates

Cartesian coordinates are a set of values that uniquely identify a point in the plane using two numbers, usually referred to as x and y. They are fundamental in defining shapes and regions in two-dimensional space.
In situations like this exercise, Cartesian coordinates help us plot and visualize equations like \(y = x^2\) and lines such as \(y = 4\). Knowing the coordinates:

• Allows you to easily locate points of intersection, which are essential for determining boundaries.
• Enables the setup of integrals with clear limits, informed by these intersections and points.
• Promotes an intuitive understanding of distance and area calculation using these plotted points.

Utilizing Cartesian coordinates effectively simplifies the process of interpreting and solving geometric problems through calculus.