Setting Up Integrals for Curves Expressed as Functions of y

Introduction

Key Concepts

1. Understanding the Area Between Curves

Finding the area between two curves involves integrating the difference between their functions over a specified interval. When the functions are expressed in terms of $y$, setting up the integral requires a different approach compared to functions of $x$. This method is essential when dealing with vertical slices or when the functions are more naturally expressed as $x$ in terms of $y$.

2. When to Use Integrals with Functions of y

Integrals with functions of $y$ are particularly useful in scenarios where:

• The region of interest is better described using horizontal slices.
• The functions defining the boundaries are easier to solve for $x$ in terms of $y$.
• The limits of integration are more naturally expressed as $y$-values.

For example, when dealing with shapes like circles or other curves where solving for $x$ in terms of $y$ simplifies the integration process, setting up integrals with respect to $y$ becomes advantageous.

3. Steps to Set Up the Integral

To set up an integral for the area between two curves expressed as functions of $y$, follow these systematic steps:

1. Identify the Curves: Determine the upper and lower functions in terms of $x$ as functions of $y$. Let's denote them as $x = f(y)$ and $x = g(y)$, where $f(y) \geq g(y)$ over the interval of interest.
2. Determine the Limits of Integration: Find the $y$-values where the two curves intersect by setting $f(y) = g(y)$ and solving for $y$. These $y$-values will serve as the lower and upper limits of the integral.
3. Set Up the Integral: The area $A$ between the curves from $y = c$ to $y = d$ is given by: $$A = \int_{c}^{d} [f(y) - g(y)] \, dy$$
4. Integrate: Perform the integration to find the area between the curves over the specified interval.

4. Example Problem

Let's consider an example to illustrate the process:

Problem: Find the area between the curves $x = y^2$ and $x = y + 2$.

Solution:

1. Identify the Curves: $f(y) = y + 2$ and $g(y) = y^2$.
2. Determine Intersection Points: Set $y + 2 = y^2$: $$y^2 - y - 2 = 0$$ $$y = \frac{1 \pm \sqrt{1 + 8}}{2} = \frac{1 \pm 3}{2}$$ $$y = 2 \quad \text{and} \quad y = -1$$
3. Set Up the Integral: $$A = \int_{-1}^{2} [(y + 2) - y^2] \, dy$$
4. Integrate: $$A = \int_{-1}^{2} (-y^2 + y + 2) \, dy$$ $$= \left[ -\frac{y^3}{3} + \frac{y^2}{2} + 2y \right]_{-1}^{2}$$ $$= \left( -\frac{8}{3} + 2 + 4 \right) - \left( \frac{1}{3} + \frac{1}{2} - 2 \right)$$ $$= \left( -\frac{8}{3} + 6 \right) - \left( \frac{1}{3} + \frac{1}{2} - 2 \right)$$ $$= \left( \frac{10}{3} \right) - \left( -\frac{7}{6} \right)$$ $$= \frac{10}{3} + \frac{7}{6} = \frac{27}{6} = 4.5$$
5. Conclusion: The area between the curves is $4.5$ square units.

5. Sketching the Region

Visualizing the region whose area is to be calculated is vital. Sketching the curves $x = f(y)$ and $x = g(y)$ on the $xy$-plane helps in understanding which function lies to the right (i.e., has a larger $x$-value) over the interval of integration. This ensures that $f(y) \geq g(y)$, maintaining the correctness of the integral setup.

In our example, $x = y + 2$ lies to the right of $x = y^2$ between $y = -1$ and $y = 2$, confirming our setup for the integral.

6. Handling Complex Regions

For more complex regions where multiple intersections occur or where one function crosses another within the interval of integration, the area calculation might require splitting the integral into multiple parts. Each part should be integrated separately over the sub-intervals where the relative positions of the functions remain consistent.

For instance, if $f(y)$ is greater than $g(y)$ from $y = a$ to $y = b$, and $g(y)$ becomes greater from $y = b$ to $y = c$, the total area is: $$A = \int_{a}^{b} [f(y) - g(y)] \, dy + \int_{b}^{c} [g(y) - f(y)] \, dy$$

7. Applications in Real-World Problems

Setting up integrals for curves expressed as functions of $y$ is not just an academic exercise; it has practical applications in fields such as engineering, physics, and economics. For example:

• Engineering: Calculating the area of cross-sections in design and manufacturing processes.
• Physics: Determining the work done by a variable force along a path.
• Economics: Finding consumer and producer surplus between supply and demand curves.

8. Integration Techniques

When dealing with integrals in terms of $y$, familiar integration techniques still apply. Depending on the complexity of $f(y)$ and $g(y)$, methods such as substitution, integration by parts, or numerical integration may be necessary. It's essential to simplify the integrand whenever possible to make the integration process manageable.

9. Common Mistakes to Avoid

Several pitfalls can occur when setting up integrals for areas between curves expressed as functions of $y$:

• Incorrect Limits of Integration: Ensure that the limits correspond to the correct $y$-values where the curves intersect.
• Misidentifying the Right Function: Always verify which function lies to the right over the interval to set up the integrand correctly.
• Algebraic Errors: Carefully solve for intersection points and simplify the integrand to prevent calculation mistakes.

10. Practice Problems

To reinforce the concepts, consider practicing with the following problems:

1. Find the area between the curves $x = \sqrt{y}$ and $x = y + 1$.
2. Determine the area enclosed by $x = y^3$ and $x = y^2$.
3. Calculate the area between $x = 4 - y^2$ and $x = y + 1$ for $y$ between $-1$ and $2$.

Attempting these problems will enhance your understanding and proficiency in setting up and evaluating integrals for areas between curves expressed as functions of $y$.

Comparison Table

Summary and Key Takeaways