Question

Describe a situation where it is advantageous to find an area enclosed by curves through integration with respect to \(y\) instead of \(x\).

Short answer

Integrating with respect to \(y\) is advantageous if curves are vertical or only easily described by \(x = f(y)\), like parabola \(x = y^2\).

Step-by-step solution

Identify the Curves

Consider a situation where you have two curves, such as a parabola opening left/right, described by a function like \(x = f(y)\), and a vertical line or another curve also in terms of \(x = g(y)\). These curves enclose an area in the plane.

Vertical Oriented Functions

When the curves are better described as \(x = f(y)\) rather than \(y = f(x)\), especially if they are vertical, it becomes complex or impossible to describe the curves in terms of \(x\). For instance, the parabola \(x = y^2\) opens sideways.

Determine Bounds

Determine the intersection points of these functions in terms of \(y\). This gives you the bounds for integration \(c\) and \(d\) to find the area between these curves.

Set Up the Integral

For the enclosed area, the expression is \( \int_{c}^{d} (g(y) - f(y)) \, dy \), where \(g(y)\) and \(f(y)\) are the respective functions describing the left and right curves.

Integrate with Respect to y

Perform the integral of \(g(y) - f(y)\) over the bounds \(c\) to \(d\). This yields the area between the curves as integrated with respect to \(y\). For instance, if \(g(y) = 4\) and \(f(y) = y^2\), you may solve \( \int_{-2}^{2} (4 - y^2) \, dy\).

Key concepts

Area Between Curves

Finding the area between curves using integration is a powerful mathematical tool, especially when curves are oriented vertically in the plane. This technique can be used when curves enclosed by either horizontal or vertical boundaries form a specific shape.
Instead of considering regions bounded by rectangular shapes, we are focusing here on more irregular shapes defined by mathematical functions. This becomes especially useful when dealing with functions that may not easily align with the horizontal axis.
To find such an area, we look for the difference between the two functions that define the boundaries of the shape at any given point. By integrating this difference over a specified interval, we can calculate the total area. However, choosing the correct axis of integration based on the curve orientation is key to simplifying this process and obtaining accurate results.

Integration Bounds

Integration bounds are crucial as they define where to start and stop calculating the area between curves. In the context of vertically oriented functions, such as sideways parabolas like the one given by the equation \(x = y^2\), these bounds are determined by the points where the curves intersect on the vertical axis.
To find these bounds, one typically sets the equations of the curves equal to one another, solving for \(y\) in lieu of \(x\). These solutions give you the values at which the functions meet or cross each other.

• Begin by solving \(f(y) = g(y)\) for \(y\).
• The solutions to the equations become the limits of integration, \(c\) and \(d\), for the integral \(\int_{c}^{d}(g(y) - f(y)) \, dy\).

Determining precise bounds ensures that the entire region is accurately calculated.

Vertical Orientation of Functions

When curves are expressed in terms of \(x = f(y)\) rather than \(y = f(x)\), the vertical orientation approach is valuable. This is common with sideways parabolas and vertical lines or expressions that open horizontally.
Attempting to describe such curves horizontally can make integration cumbersome or even infeasible. Integration with respect to \(y\) simplifies the process because we're taking slices from top to bottom rather than left to right.

• This is particularly helpful when the function is not easily invertible into the form \(y = f(x)\).
• Utilizing the form \(x = f(y)\) allows more natural handling of vertical curves.

Recognizing when to switch perspectives in integration is critical for easier calculation and understanding of the curve behaviors.

Sideways Parabolas

Sideways parabolas, unlike their upright counterparts, open horizontally. This unique orientation often necessitates the use of integration with respect to \(y\).
An example is the function \(x = y^2\), which cannot be expressed as a simple function of \(x\) in a \(y = f(x)\) form. Instead, it's best handled by treating \(x\) as a function of \(y\).

• These shapes stretch horizontally rather than vertically, making calculations with traditional horizontal slices difficult.
• Integration with respect to \(y\) sees the curves split left and right at each \(y\) value, which simplifies the area calculation.

Understanding the orientation of the parabola is key in selecting the right method for area calculations, ensuring the accuracy and simplicity of your work.