Problem 3 - University Calculus: Early Transcendentals

Question

Sketch the described regions of integration. $$-2 \leq y \leq 2, \quad y^{2} \leq x \leq 4$$

Step-by-Step Solution

Answer

The region is between the parabola \(x = y^2\) and vertical line \(x = 4\), for \(-2 \leq y \leq 2\).

Step 1: Understanding the Boundaries for y

The problem states that the value of y is bounded by \(-2 \leq y \leq 2\).This implies that y can take on any value between -2 and 2 inclusive, forming vertical limits for the region.

Step 2: Understanding the Boundaries for x

The condition for x is given as\(y^2 \leq x \leq 4\).This means, for a given y, x is confined between the square of y and 4. The lower boundary \(x = y^2\) represents a parabola opening to the right, and the upper boundary \(x = 4\) is a vertical line.

Step 3: Sketch the Boundaries

Start by sketching the parabola \(x = y^2\) between y = -2 and y = 2. This parabola opens to the right and passes through points such as (4, 2), (0, 0), and (4, -2). Then, sketch the vertical line \(x = 4\) which is constant for all y between -2 and 2.

Step 4: Identify the Region of Integration

The region is bounded on the left by \(x = y^2\) and on the right by \(x = 4\). Vertically, it stretches from y = -2 to y = 2. Shade the area between the parabola and the vertical line from y = -2 up to y = 2 to represent the region of integration.

Key Concepts

Region of Integration

In calculus, identifying the region of integration is crucial for solving multiple integrals. This region is the area over which the integral is calculated. In our example, the region of integration is confined between the boundaries provided by the functions in both the x and y directions.

Understanding the region's shape is essential. In the given problem, it is delineated by the parabola and a line that guides where the integration occurs. It looks a bit like a slice bounded by these curves.

When sketching the region of integration:

• Start by identifying each boundary.
• Sketch the curves or lines to see where they meet or intersect.
• Shade or mark the area that represents the bounded region.

Visualizing this region helps in setting up the correct limits for integrating the given functions.

Parabola

A parabola is a U-shaped curve described by quadratic equations like \(x = y^2\). In this problem, the parabola opens to the right, which is typical when the parabola is defined for x in terms of y.

This specific parabola is significant because it sets one of the integration boundaries. The points where it intersects y-axis give key limits in our region of integration such as (0, 0).

• Observe that as \(y\) changes from -2 to 2, the parabola sweeps across horizontally.
• It acts as the leftmost boundary for our integrated area within the given region.

Understanding the parabola's behavior helps to correctly establish limits of integration for the integral's evaluation.

Vertical Limits

The vertical limits refer to the range of y-values that encapsulate the region of integration. Here, the problem specifies the range as \( −2 \leq y \leq 2 \). These y-values determine the topmost and bottommost edges of our area; they frame the vertical span of our region. The limits are clear-cut:

• They dictate that any slice we take horizontally will have y fall within this numerical range.
• In practical terms, this means the region extends vertically between -2 to +2 on the y-axis.

Making sense of the vertical limits simplifies developing insights about the region's total bounds, vital when performing definite integrations.

Integration Boundaries

Integration boundaries are the parameters within which the integration occurs. In the context of this exercise, these boundaries play a critical role in determining where we evaluate the integral.

Defined by equations \(y^2 \leq x \leq 4\), with \(x = y^2\) and a vertical line \(x = 4\), these marks set the confines for our area of interest. A few pointers to remember:

• Horizontal boundary is influenced by x-values expressed in relation to y-values.
• Left boundary is the curve from parabola \(x = y^2\).
• Right boundary is the vertical constant line at \(x = 4\).

Properly establishing such boundaries simplifies the task of setting up the integrals, ensuring they accurately reflect the physical region described in the problem.