Problem 8 - University Calculus: Early Transcendentals

Question

Sketch the described regions of integration. $$0 \leq y \leq 8, \quad \frac{1}{4} y \leq x \leq y^{1 / 3}$$

Step-by-Step Solution

Answer

Sketch the region between \(x = \frac{1}{4} y\) and \(x = y^{1/3}\) from \(y = 0\) to \(y = 8\).

Step 1: 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 \(y = 8\).

Step 2: 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.

Step 3: 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 \(y\).

Step 4: 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 \(y\).

Step 5: 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.

Step 6: 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 \(y = 8\).

Key Concepts

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 \((2,8)\).

Notice how these points line up in a straight line through the origin, slanting upwards with a moderate slope. Line sketching becomes very useful as this clearly shows one boundary of the region based on the inequality \(\frac{1}{4}y \leq x\).

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 \((2,8)\).

Unlike the straight line, the curve \(x = y^{1/3}\) rises quickly at first then levels off. Crucially, this curve bounds the other side of the region, forming a varied slope on a graph visually contrasting with the linear equality.

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 \(x = y^{1/3}\).

Begin by sketching the line and curve on the same axes to see how they overlay. The shaded region in between them, for every point \(y\) from 0 to 8, demonstrates where those inequalities are simultaneously satisfied. This approach ensures a clear, visual cutting guide through potentially complex algebraic boundaries, aligning and clarifying their impact on real-world integrations.