📚 Topic Summary
Finding the area between curves when the equations are given as functions of $y$ (i.e., $x = f(y)$ and $x = g(y)$) involves integrating with respect to $y$. The area $A$ is given by the integral of the absolute difference between the two functions over the interval $[c, d]$, where $c$ and $d$ are the $y$-values of the intersection points. This means $A = \int_{c}^{d} |f(y) - g(y)| dy$. To find the area, you need to identify the curves, find the intersection points, and then set up and evaluate the definite integral. Remember to subtract the left function from the right function.
The basic concept is finding where the curves intersect and then integrating the difference of the functions along the y-axis.
🧮 Part A: Vocabulary
Match the term with its definition:
- Definite Integral
- Intersection Points
- Area Between Curves
- Function of y
- Integration Limits
- The y-values where two curves meet.
- A function where x is expressed in terms of y (x = f(y)).
- The numerical value representing the area under a curve between two points.
- The y-values that define the start and end of the integration.
- The region enclosed by two or more curves.
(Match the numbers with the letters to test your knowledge!)
✍️ Part B: Fill in the Blanks
The area between curves defined as functions of $y$ is found by integrating with respect to ____. We need to find the ____ points of the curves to determine the limits of ____. The integral is set up by subtracting the ____ function from the ____ function, all with respect to $y$. The result of this integral gives us the ____ between the curves.
🤔 Part C: Critical Thinking
Explain, in your own words, why it's important to consider the orientation of the curves (which is to the right and which is to the left) when calculating the area between curves defined by y-functions. What happens if you don't account for this correctly?