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Chapter 14: Problem 13

Sketch the following regions and write an iterated integral of a continuous function \(f\) over the region. Use the order \(d y d x\) \(R\) is the triangular region with vertices \((0,0),(0,2),\) and (1,0)

Short Answer

Expert verified
Answer: \(\int_{0}^{1} \int_{0}^{-2x+2} f(x,y)\,\mathrm{d}y\,\mathrm{d}x\)

Step by step solution

01

Sketch the Region

To sketch the triangular region \(R\), we plot the three vertices \((0, 0)\), \((0, 2)\), and \((1, 0)\). Then, connect these vertices to form a triangle. The three sides of the triangle are given by the equations: 1. \(y = 0\) 2. \(x = 0\) 3. \(y = -2x + 2\) So, the region \(R\) is bounded by these three lines.
02

Write the Iterated Integral

To write an iterated integral, we first find the limits of integration for \(x\) and \(y\). From the triangular region \(R\), we can see that: - \(x\) varies from \(0\) to \(1\) - For each value of \(x\), \(y\) varies from \(0\) to \(-2x + 2\) Now, we can write the iterated integral of continuous function \(f\) over region \(R\) in the order of $\displaystyle\mathrm{d}y\mathrm{d}x: \begin{align*} \int_{0}^{1} \int_{0}^{-2x+2} f(x,y)\,\mathrm{d}y\,\mathrm{d}x \end{align*} This is the iterated integral of continuous function \(f\) over the triangular region \(R\) with the given order \(\displaystyle\mathrm{d}y\mathrm{d}x\).

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Key Concepts

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

Sketching Regions
Visualizing a region is a crucial first step in setting up an iterated integral. In this problem, we're dealing with a triangular region, which can be challenging but rewarding to sketch correctly.
This specific triangle has vertices at \(0, 0\), \(0, 2\), and \(1, 0\). To sketch it, start by plotting these points on a coordinate plane. Once plotted, connect them with straight lines.
The boundaries of this triangle are determined by three line equations: *y = 0*, which is the x-axis; *x = 0*, which is the y-axis; and *y = -2x + 2*, which forms the hypotenuse of the triangle. By sketching this triangle, you better understand the domain over which you'll integrate.
Limits of Integration
Understanding the limits of integration is key to setting up an iterated integral. For the given triangular region, you need to determine how the variables range. This involves identifying how the boundaries influence the limits.
In this case, you'll work from left to right and from bottom to top. \(x\) ranges from 0 to 1, as these are the horizontal extents of the triangle from vertex \(0, 0\) to \(1, 0\). Meanwhile, \(y\) depends on the \(x\)-coordinate in a more complex way, varying between 0 and -2x + 2, which slides linearly down from the top to the left side of the triangle.
  • The lower limit of \(x\) is 0 and the upper limit is 1.
  • For each \(x\), \(y\) begins at 0 and extends to -2x + 2.
This systematic approach ensures you correctly establish the integral's bounds, effectively allowing you to evaluate it accurately.
Continuous Functions
Continuous functions are essential in calculus, particularly in integration. When dealing with iterated integrals, the continuity of the function \(f(x, y)\) ensures that the integration is well-defined over the entire triangular region.
Such functions, which don't break or jump, allow for the use of simple integration techniques. When integrating iteratively, you perform integration with respect to one variable, and then the next, without complication from discontinuities.
In our exercise, the iterated integral is given as: \[ \int_{0}^{1} \int_{0}^{-2x+2} f(x,y)\,\mathrm{d}y\,\mathrm{d}x \]This provides an effective framework for calculating the total effect of \(f\) over the region. By keeping \(f\) continuous, we can confidently apply these limits of integration and trust the process will yield a meaningful result.

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Most popular questions from this chapter

Use integration to find the volume of the following solids. In each case, choose a convenient coordinate system, find equations for the bounding surfaces, set up a triple integral, and evaluate the integral. Assume that \(a, b, c, r, R,\) and h are positive constants. Find the volume of a solid right circular cone with height \(h\) and base radius \(r\).

The Jacobian is a magnification (or reduction) factor that relates the area of a small region near the point \((u, v)\) to the area of the image of that region near the point \((x, y)\) a. Suppose \(S\) is a rectangle in the \(u v\) -plane with vertices \(O(0,0)\) \(P(\Delta u, 0),(\Delta u, \Delta v),\) and \(Q(0, \Delta v)\) (see figure). The image of \(S\) under the transformation \(x=g(u, v), y=h(u, v)\) is a region \(R\) in the \(x y\) -plane. Let \(O^{\prime}, P^{\prime},\) and \(Q^{\prime}\) be the images of O, P, and \(Q,\) respectively, in the \(x y\) -plane, where \(O^{\prime}\) \(P^{\prime},\) and \(Q^{\prime}\) do not all lie on the same line. Explain why the coordinates of \(\boldsymbol{O}^{\prime}, \boldsymbol{P}^{\prime}\), and \(Q^{\prime}\) are \((g(0,0), h(0,0))\) \((g(\Delta u, 0), h(\Delta u, 0)),\) and \((g(0, \Delta v), h(0, \Delta v)),\) respectively. b. Use a Taylor series in both variables to show that $$\begin{aligned} &g(\Delta u, 0) \approx g(0,0)+g_{u}(0,0) \Delta u\\\ &g(0, \Delta v) \approx g(0,0)+g_{v}(0,0) \Delta v\\\ &\begin{array}{l} h(\Delta u, 0) \approx h(0,0)+h_{u}(0,0) \Delta u \\ h(0, \Delta v) \approx h(0,0)+h_{v}(0,0) \Delta v \end{array} \end{aligned}$$ where \(g_{u}(0,0)\) is \(\frac{\partial x}{\partial u}\) evaluated at \((0,0),\) with similar meanings for \(g_{v}, h_{u}\) and \(h_{v}\) c. Consider the vectors \(\overrightarrow{O^{\prime} P^{\prime}}\) and \(\overrightarrow{O^{\prime} Q^{\prime}}\) and the parallelogram, two of whose sides are \(\overrightarrow{O^{\prime} P^{\prime}}\) and \(\overrightarrow{O^{\prime} Q^{\prime}}\). Use the cross product to show that the area of the parallelogram is approximately \(|J(u, v)| \Delta u \Delta v\) d. Explain why the ratio of the area of \(R\) to the area of \(S\) is approximately \(|J(u, v)|\)

Let \(D\) be the region bounded by the ellipsoid \(x^{2} / a^{2}+y^{2} / b^{2}+z^{2} / c^{2}=1,\) where \(a > 0, b > 0\) and \(c>0\) are real numbers. Let \(T\) be the transformation \(x=a u, y=b v, z=c w\). Evaluate \(\iiint_{D}|x y z| d A\)

Compute the volume of the following solids. Wedge The wedge sliced from the cylinder \(x^{2}+y^{2}=1\) by the planes \(z=a(2-x)\) and \(z=a(x-2),\) where \(a>0\)

Find the volume of the four-dimensional pyramid bounded by \(w+x+y+z+1=0\) and the coordinate planes \(w=0, x=0, y=0,\) and \(z=0\).
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