Problem 5 - University Calculus: Early Transcendentals

Question

Sketch the described regions of integration. $$0 \leq x \leq 1, \quad e^{x} \leq y \leq e$$

Step-by-Step Solution

Answer

Shade the region where \( 0 \leq x \leq 1 \) and \( e^x \leq y \leq e \).

Step 1: Understand the region of integration

We have two inequalities to describe the region in the xy-plane: one for x, which is that it ranges from 0 to 1, and one for y, which ranges from \( e^x \) to \( e \). The first inequality specifies the x-boundaries and the second specifies the y-boundaries.

Step 2: Identify x-boundaries

The inequality \( 0 \leq x \leq 1 \) tells us that x starts at 0 and ends at 1. This describes a vertical slice of the plane that is one unit wide.

Step 3: Determine y-boundaries

For each x in the range from 0 to 1, y ranges from \( e^x \) to \( e \). This means the lower boundary is the curve \( y = e^x \) and the upper boundary is the horizontal line \( y = e \).

Step 4: Sketch the boundaries

Draw the line \( x = 0 \) and \( x = 1 \) on the xy-plane. Then sketch the curve \( y = e^x \) starting at the point (0,1) rising as x increases, and the horizontal line \( y = e \), which intersects the y-axis at a height of approximately 2.718.

Step 5: Highlight the region

Shade the region between x=0 and x=1, and between the curve \( y = e^x \) and the line \( y = e \). This is the area of integration as defined by the given boundaries.

Key Concepts

Inequalities

In calculus, inequalities play a crucial role in defining a region of integration on a graph. They are used to specify the ranges of variables, usually x and y, that form the boundaries of an area. For instance, if given the inequalities \(0 \leq x \leq 1\) and \(e^x \leq y \leq e\), these expressions set constraints for x and y, creating a confined area on the xy-plane.

• The inequality \(0 \leq x \leq 1\) means that x can take any value between 0 and 1, inclusive.
• The inequality \(e^x \leq y \leq e\) means that for each value of x, y can fluctuate from the curve \(y = e^x\) up to the horizontal line \(y = e\).

Understanding these inequalities is essential for visualizing the region of integration. They tell us exactly how x and y are bounded and help in determining how to shade the correct area on the graph.

Sketching Regions

When sketching regions of integration, it’s important to plot the graph accurately, respecting the boundaries set by the inequalities. Begin by plotting any curves and lines described by the equations in the inequalities. Here’s a breakdown to help visualize:

• Draw vertical lines at \(x = 0\) and \(x = 1\) to indicate where x is constrained.
• Sketch the curve \(y = e^x\), starting from the point (0,1). This curve rises smoothly as x increases.
• Plot the horizontal line \(y = e\), which typically intersects the y-axis at approximately 2.718.

Once you have drawn the necessary lines and curves, shade the area between them. This visual representation helps in comprehending the region that is defined by the inequalities, setting the stage for integration.

Integration Boundaries

Integration boundaries define the limits for evaluating definite integrals. In a region of integration, boundaries like \(0 \leq x \leq 1\) and \(e^x \leq y \leq e\) determine where the integration begins and ends for each variable.

• The intervals \(0 \leq x \leq 1\) set the limits for the variable x. This means when performing integration with respect to x, these are the points where the integration starts and stops.
• Every slice of x from 0 to 1 has a y-bound that ranges between \(y = e^x\) and \(y = e\). These are the limits of integration for y for any given fixed x.

Integrating over this region involves using these boundaries to set up the integral. This process filters out the area enclosed by the boundaries and allows for the evaluation of the integral within the specified limits.