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Chapter 16: Problem 6

Sketch the region of integration for \(\int_{-2}^{2} \int_{x^{2}}^{4} e^{x} d y d x\)

Short Answer

Expert verified
The region of integration is a closed area bounded by the parabola \(y = x^2\), the horizontal line y = 4, and the vertical lines x = -2 and x = 2.

Step by step solution

01

Identify the limits of integration

The given double integral is: \(\int_{-2}^{2} \int_{x^{2}}^{4} e^{x} d y d x\). Here, the limits of integration are -2 and 2 for x, and \(x^2\) and 4 for y.
02

Determine the equations for the boundaries

Now that we have identified the limits of integration, we can write the equations for the boundaries of the region. From the given limits, we can deduce: 1. x varies from -2 to 2, which are two vertical lines: x = -2 and x = 2. 2. y varies from \(x^2\) to 4. The lower bound is \(y = x^2\), which is a parabola opening upwards with its vertex at the origin. The upper bound is y = 4, which is a horizontal line.
03

Sketch the region of integration

We will now sketch the identified boundaries to form the region of integration. Plot the two vertical lines x = -2 and x = 2, the parabola \(y = x^2\), and the horizontal line y = 4. The region of integration is the area enclosed by these lines. The region of integration is a closed area bounded by the parabola \(y = x^2\), the horizontal line y = 4 and the vertical lines x = -2 and x = 2.

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Key Concepts

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

Region of Integration
When tackling double integrals, one essential concept is identifying the **region of integration**. This region is the area over which the integration is performed.
In the example, through the given limits, the region is defined in the xy-plane.
The boundaries we need to consider involve:
  • Two vertical lines: x = -2 and x = 2.
  • A parabola described by y = x虏, which opens upwards.
  • A horizontal line y = 4.
Together, these elements create a confined area. Essentially, this region is where x and y values vary according to the limits given. Understanding this helps set up the problem visually and mathematically.
Limits of Integration
**Limits of Integration** are the values that define where the boundaries of the region begin and end. They are crucial for solving any integral effectively. The integral \[ \int_{-2}^{2} \int_{x^{2}}^{4} e^{x} \, dy \, dx \] provides us with specific limits:
  • The limit for x is from -2 to 2.
  • The limit for y is from \(x^2\) to 4.
These limits tell us how the integration is confined in space, guiding us on what section of the graph we're analyzing.
They provide the boundary conditions for solving the problem, and any incorrect interpretation could lead to incorrect solutions. Always ensure that you thoroughly interpret these limits to match the region correctly.
Boundaries of Integration
The **Boundaries of Integration** give visual insight into the problem. They are literally the borders that enclose the region where integration takes place.
For the given integral, the boundaries can be visualized as follows:
  • The line x = -2 is a vertical boundary on the left.
  • The line x = 2 is a vertical boundary on the right.
  • The parabola y = x虏 forms the bottom boundary.
  • The line y = 4 serves as the top horizontal boundary.
These boundaries can help us visualize the shape, which is crucial for understanding the integral.
They form a closed area, ensuring our region is well-defined and ready for integration. Visualizing these boundaries on a graph will always assist in a clearer understanding of the problem.

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Most popular questions from this chapter

Write iterated integrals in spherical coordinates for the following regions in the specified orders. Sketch the region of integration. Assume \(g\) is continuous on the region. \(\int_{0}^{2 \pi} \int_{0}^{\pi / 2} \int_{0}^{4 \sec \varphi} g(\rho, \varphi, \theta) \rho^{2} \sin \varphi d \rho d \varphi d \theta\) in the orders \(d \rho d \theta d \varphi\) and \(d \theta\) d\rho \(d \varphi\).

Evaluate the following integrals using polar coordinates. Assume \((r, \theta)\) are polar coordinates. A sketch is helpful. $$\int_{0}^{3} \int_{0}^{\sqrt{9-x^{2}}} \sqrt{x^{2}+y^{2}} d y d x$$

Intersecting spheres One sphere is centered at the origin and has a radius of \(R\). Another sphere is centered at \((0.0, r)\) and has a radius of \(r,\) where \(r>R / 2 .\) What is the volume of the region common to the two spheres?

Evaluate the following integrals using polar coordinates. Assume \((r, \theta)\) are polar coordinates. A sketch is helpful. $$\iint_{R}\left(x^{2}+y^{2}\right) d A ; R=\\{(r, \theta): 0 \leq r \leq 4,0 \leq \theta \leq 2 \pi\\}$$

Find the volume of the solid bounded by the surface \(z=f(x, y)\) and the \(x y\)-plane. (Check your book to see figure) $$f(x, y)=16-4\left(x^{2}+y^{2}\right)$$
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