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Magazine Chapter 14: Problem 8
Sketch the described regions of integration. $$0 \leq y \leq 8, \quad \frac{1}{4} y \leq x \leq y^{1 / 3}$$
Short Answer
Expert verified
Sketch the region between \(x = \frac{1}{4} y\) and \(x = y^{1/3}\) from \(y = 0\) to \(y = 8\).
Step by step solution
01
Understand the Boundaries for y
The inequality \(0 \leq y \leq 8\) indicates that \(y\) can range from 0 to 8. This tells us that the region will extend vertically from \(y = 0\) to \(y = 8\).
02
Understand the Boundaries for x
The inequalities \(\frac{1}{4} y \leq x \leq y^{1/3}\) provide the bounds for \(x\) in terms of \(y\). As \(y\) varies between 0 and 8, \(x\) ranges between \(x = \frac{1}{4}y\) and \(x = y^{1/3}\). This means that for each \(y\) value, \(x\) will be constrained between these two bounds.
03
Sketch the line x = y/4
Plot the line \(x = \frac{1}{4} y\) by noting that it passes through points like \((0,0), (2,8), (1,4)\), which is a straight line through the origin with a shallow slope. This line represents the lower boundary for \(x\) in terms of \(y\).
04
Sketch the curve x = y^{1/3}
Plot the curve \(x = y^{1/3}\) by noting characteristic points like \((0,0), (1,1), (8,2)\). This curve rises more rapidly than the line \(x = \frac{1}{4} y\) because it involves a root function of \(y\).
05
Identify the Region of Integration
The region of integration lies between the line \(x = \frac{1}{4} y\) and the curve \(x = y^{1/3}\) for all \(y\) values from 0 to 8. Shade this region in your sketch.
06
Finalize the Sketch
Ensure the sketch clearly shows the region between the line and curve, for \(0 \leq y \leq 8\). The shaded area represents the region of integration, bounded horizontally by \(x = \frac{1}{4} y\) and \(x = y^{1/3}\), and vertically by \(y = 0\) and \(y = 8\).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Line Sketching
Line sketching involves creating a visual representation of linear equations, using their algebraic forms to plot corresponding points on a graph. In the exercise, we're interested in sketching the line represented by the equation \(x = \frac{1}{4}y\). To do this, we choose several values of \(y\) and calculate corresponding \(x\) values using \(x = \frac{1}{4}y\). Some simple pairs include \((0,0), (4,1), (8,2)\), and so on.
- Begin at \((0,0)\), where both \(x\) and \(y\) are zero.
- Next, for \(y = 4\), \(x\) is \(1\), so plot the point \((1,4)\).
- For \(y = 8\), \(x\) becomes \(2\), giving the point \((2,8)\).
Curve Sketching
Curve sketching is similar to line sketching but focuses on non-linear equations. Here, we're sketching the curve \(x = y^{1/3}\). We will again choose values for \(y\) and determine corresponding \(x\) values by solving \(x = y^{1/3}\). This results in pairs such as \((0,0), (1,1), (8,2)\).
- All values start at \((0,0)\), where both \(x\) and \(y\) are zero.
- For \(y = 1\), \(x\) equals \(1\), resulting in \((1,1)\).
- At \(y = 8\), \(x\) becomes \(2\), providing the point \((2,8)\).
Inequalities in Integration
Understanding inequalities in integration is fundamental when identifying the region of integration on a graph. In the given exercise, two inequalities govern this process: \( \frac{1}{4}y \leq x \leq y^{1/3} \) within \(0 \leq y \leq 8\).
- The first inequality \(\frac{1}{4}y \leq x\) sets the lower bound for \(x\), visualized by the line \(x=\frac{1}{4}y\).
- The second inequality \(x \leq y^{1/3}\) specifies the upper limit, interpreted as the curve \(x = y^{1/3}\).
