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Chapter 13: Problem 37

Write the given iterated integral as an iterated integral with the order of integration interchanged. Hint: Begin by sketching a region \(S\) and representing it in two ways, as in Example \(5 .\) $$ \int_{0}^{1} \int_{-y}^{y} f(x, y) d x d y $$

Short Answer

Expert verified
The reordered integral is \( \int_{-1}^{1} \int_{0}^{|x|} f(x, y) \ dy \ dx \).

Step by step solution

01

Identify the given order of integration limits

The given iterated integral is \( \int_{0}^{1} \int_{-y}^{y} f(x, y) \ dx \ dy \). The limits for \(x\) are from \(-y\) to \(y\), and the limits for \(y\) are from \(0\) to \(1\).
02

Sketch the region of integration

To identify the region \(S\), sketch the lines \(x = -y\), \(x = y\), \(y = 0\), and \(y = 1\). The intersection of these boundaries forms a triangular region in the xy-plane.
03

Describe the region as a type II region

In the xy-plane, the region is bounded by \(y \geq 0\), \(y \leq 1\), \(x \geq -y\), and \(x \leq y\). This region is described in terms of \(y\) as a function of \(x\) for changing the order of integration.
04

Rearrange the bounds for new order of integration

To integrate first with respect to \(y\), determine the limits based on the region. \(y\) ranges from \(-x\) to \(x\), while \(x\) ranges from \(0\) to \(1\).
05

Write the iterated integral with interchanged order

The iterated integral with interchanged order is \( \int_{-1}^{1} \int_{0}^{|x|} f(x, y) \ dy \ dx \). The region is now defined with respect to \(x\).

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Key Concepts

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

Order of Integration
In iterated integrals, the order of integration refers to the sequence in which the integral is taken with respect to each variable. For a double integral in the form \( \int \int f(x, y) \, dx \, dy \), it means integrating with respect to \(x\) first, followed by \(y\). Sometimes, it might be necessary to change the order of integration. This can simplify the problem or make the integration possible.
The original problem starts with integrating \(x\) from \(-y\) to \(y\), and then \(y\) from 0 to 1. After swapping, we integrate \(y\) first over \([-x, x]\) and then \(x\) from 0 to 1.
Region of Integration
The region of integration is the area over which the function is being integrated. It is crucial to first identify this region in the xy-plane, which helps us determine the limits for integration.
In the example, the specified region \(S\) is defined by:
  • \(y \geq 0\)
  • \(y \leq 1\)
  • \(x \geq -y\)
  • \(x \leq y\)
This forms a triangular shape in the xy-plane, bounded by the lines \(x = -y\), \(x = y\), \(y = 0\), and \(y = 1\). Understanding this region allows us to correctly change the order of integration.
Sketching Regions
Sketching the region of integration is a helpful visual tool that makes it easier to rearrange the bounds for integration. By plotting the boundaries \(x = -y\), \(x = y\), \(y = 0\), and \(y = 1\), you can clearly see the triangular region in the xy-plane. Sketching is essential because:
  • It visually verifies the boundaries and their intersections.
  • Helps determine new bounds when changing the order of integration.
  • Ensures the region of integration is correctly represented.
Making an accurate sketch is a crucial step before attempting to change the order of integration in an iterated integral.
Integration Limits
Integration limits are the values that define the start and end points for the integral. These limits must be carefully analyzed, especially when the order of integration changes.
For the initial integral \(\int_{0}^{1} \int_{-y}^{y} f(x,y) \, dx \, dy\), the limits depend on the variables \(x\) and \(y\). After changing the order, the limits are identified as \( \int_{-1}^{1} \int_{0}^{|x|} f(x,y) \, dy \, dx\). Here:
  • \(y\) is bounded by \([-x, x]\) when changing the integration order.
  • \(x\) ranges from 0 to 1.
Correctly determining these limits ensures that the integration is performed over the correct region.

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Most popular questions from this chapter

Find the moments of inertia \(I_{x}, I_{y}\), and \(I_{z}\) for the lamina bounded by the given curves and with the indicated density \(\delta .\) Triangle with vertices \((0,0),(0, a),(a, 0) ; \delta(x, y)=\) \(x^{2}+y^{2}\)

Parallel Axis Theorem Consider a lamina \(S\) of mass \(m\) together with parallel lines \(L\) and \(L^{\prime}\) in the plane of \(S\), the line \(L\) passing through the center of mass of \(S\). Show that if \(I\) and \(I^{\prime}\) are the moments of inertia of \(S\) about \(L\) and \(L^{\prime}\), respectively, then \(I^{\prime}=I+d^{2} m\), where \(d\) is the distance between \(L\) and \(L^{\prime}\). Hint: Assume that \(S\) lies in the \(x y\) -plane, \(L\) is the \(y\) -axis, and \(L^{\prime}\) is the line \(x=-d\).

Six surfaces are given below. Without performing any integration, rank the surfaces in order of their surface area from smallest to largest. Hint: There may be some "ties." (a) The paraboloid \(z=x^{2}+y^{2}\) over the region in the first quadrant and inside the circle \(x^{2}+y^{2}=1\) (b) The hyperbolic paraboloid \(z=x^{2}-y^{2}\) over the region in the first quadrant and inside the circle \(x^{2}+y^{2}=1\) (c) The paraboloid \(z=x^{2}+y^{2}\) over the region inside the rectangle with vertices \((0,0),(1,0),(1,1)\), and \((0,1)\) (d) The hyperbolic paraboloid \(z=x^{2}-y^{2}\) over the region inside the rectangle with vertices \((0,0),(1,0),(1,1)\), and \((0,1)\) (e) The paraboloid \(z=x^{2}+y^{2}\) over the region inside the triangle with vertices \((0,0),(1,0)\), and \((0,1)\) (f) The hyperbolic paraboloid \(z=x^{2}-y^{2}\) over the region inside the triangle with vertices \((0,0),(1,0)\), and \((0,1)\)

Find the Jacobian for the transformation from rectangular coordinates to spherical coordinates.

Find the area of the indicated surface. Make a sketch in each case. The part of the saddle \(a z=x^{2}-y^{2}\) inside the cylinder \(x^{2}+y^{2}=a^{2}, a>0\).
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