Problem 12 - Advanced Engineering Mathematics

Question

Sketch the region of integration for the given iterated integral. $$ \int^{2} \int^{x^{2}+1} f(x, y) d y d x $$

Step-by-Step Solution

Answer

The region is bounded by \( y = x^2 + 1 \), \( y = 4 \), and \( x = 0 \) to \( x = 2 \).

Step 1: Understand the Given Integral Limits

The given iterated integral is \( \int_{x=0}^{x=2} \int_{y=x^2+1}^{y=4} f(x, y)\, dy \, dx \). This notation indicates that integration is first performed with respect to \( y \), with \( y \) ranging from \( x^2 + 1 \) to 4, and then with respect to \( x \), ranging from \( 0 \) to \( 2 \).

Step 2: Identify the Ranges for y

The limits of \( y \) are given by \( x^2 + 1 \leq y \leq 4 \). This means for a fixed \( x \), \( y \) starts from the curve \( y = x^2 + 1 \) and goes up to the horizontal line \( y = 4 \).

Step 3: Identify the Ranges for x

The limits for \( x \) are straightforward from the outer integral as \( 0 \leq x \leq 2 \). Hence, \( x \) varies between 0 and 2 in the sketch.

Step 4: Sketch the Curves

First, sketch the parabola \( y = x^2 + 1 \). This is a vertical shift of the standard parabola \( y = x^2 \) upwards by 1 unit. Next, sketch the line \( y = 4 \), which is a horizontal line across the entire range of \( x \).

Step 5: Draw the Region of Integration

Within the plane from \( x=0 \) to \( x=2 \), shade the area between the curve \( y = x^2 + 1 \) and the line \( y = 4 \). This is the region bounded by both these lines until \( x \) reaches its limits.

Key Concepts

Integration Limits

When dealing with iterated integrals, understanding the integration limits is crucial. They indicate the boundaries over which the integration process occurs. In the given integral \( \int_{x=0}^{x=2} \int_{y=x^2+1}^{y=4} f(x, y)\, dy \, dx \), the limits are expressed as ranges for the variables involved.

• The limits for \( y \) are found in the inner integral: \( y = x^2 + 1 \) to \( y = 4 \). This tells us that for each fixed \( x \), \( y \) takes values starting from the curve \( y = x^2 + 1 \) and ending at the line \( y = 4 \).
• The limits for \( x \) are given by the outer integral: \( x = 0 \) to \( x = 2 \). This range sets the boundary for \( x \) and applies to the entire process of integrating over \( y \).

Understanding these limits is essential because they define the region of integration and tell us where the integration process takes place for each variable.

Region of Integration

The region of integration is the specific area in the coordinate plane where the integration occurs. For the iterated integral given by \( \int_{x=0}^{x=2} \int_{y=x^2+1}^{y=4} f(x, y)\, dy \, dx \), this region is bounded by different curves and lines:

• This area is defined horizontally by the values of \( x \), which range from \( 0 \) to \( 2 \). Vertical boundaries come from the limits of \( y \), which go from the curve \( y = x^2 + 1 \) up to the line \( y = 4 \).
• The combination of these boundaries forms a specific region that we need to carefully visualize and understand. It's a section of the coordinate plane trapped between a quadratic curve (a parabola) and a straight line, exclusively within the \( x \) boundaries.

This region of integration explains where the integral accumulates its values and helps in visualizing how changes in \( x \) and \( y \) affect the area being considered.

Sketching Curves

Sketching the curves involved in an iterated integral helps visualize the region of integration. Let's break down how to sketch the relevant curves for our particular integral:

• Parabola \( y = x^2 + 1 \): Start by sketching a standard parabola \( y = x^2 \), which passes through the origin. Shift this entire curve upwards by 1 unit to model the equation accurately. The result is a parabola that opens upwards, beginning at \( y = 1 \).
• Horizontal Line \( y = 4 \): This is simpler to draw. It's a straight line parallel to the x-axis, located at the height \( y = 4 \). Make sure the line extends across the entire range of \( x \) from \( 0 \) to \( 2 \).

Next, draw vertical lines to mark the region between \( x=0 \) and \( x=2 \). The shaded area between these lines shows the intersecting area between the parabola and the horizontal line, indicating the region of integration.