📚 Topic Summary
When finding the area between curves, sometimes it's easier to express the curves as functions of $y$ rather than $x$. This is especially helpful when the curves intersect more than once, or when the functions are easier to integrate with respect to $y$. Instead of integrating 'top minus bottom', we integrate 'right minus left'. The area $A$ between the curves $x = f(y)$ and $x = g(y)$ from $y = c$ to $y = d$, where $f(y) \geq g(y)$ on the interval $[c, d]$, is given by the integral: $A = \int_{c}^{d} [f(y) - g(y)] dy$.
Remember to identify the interval of integration along the y-axis and ensure that $f(y)$ is always to the right of $g(y)$ within that interval.
🧠 Part A: Vocabulary
Match the terms with their definitions:
- Term: Horizontal Strip
- Term: Function of $y$
- Term: Right Boundary
- Term: Left Boundary
- Term: Integration with respect to $y$
- Definition: A function where $x$ is expressed in terms of $y$, i.e., $x = f(y)$.
- Definition: The curve farthest to the right when integrating with respect to $y$.
- Definition: Performing the integral using $dy$ as the differential.
- Definition: The curve farthest to the left when integrating with respect to $y$.
- Definition: A thin rectangle drawn horizontally between the curves to approximate the area.
Match each term (1-5) to its correct definition (a-e).
✏️ Part B: Fill in the Blanks
Complete the following paragraph with the correct words:
To find the area between curves that are functions of $y$, we integrate with respect to _____. We subtract the _____ function from the _____ function within the limits of integration along the _____-axis. The formula is given by $A = \int_{c}^{d} [f(y) - g(y)] dy$, where $f(y)$ is to the _____ of $g(y)$.
🤔 Part C: Critical Thinking
Explain why it is sometimes easier to integrate with respect to $y$ rather than $x$ when finding the area between curves. Provide an example of a scenario where integrating with respect to $y$ would be significantly simpler.