Loading the lesson
Loading the lesson
Search AllRounder.ai
Search the courses, subjects, tracks, games and features, or jump straight to a page.
Search the courses, subjects, tracks, games and features, or jump straight to a page.
Learn it your way
A short class with Sarah and her students. The other ways are under More.
Sign up free to listen. Transcripts are free to read.
Today, we're exploring how to calculate the area when the function is defined in terms of y, specifically using the function x = f(y). Can anyone tell me why we might prefer this form?
Maybe because sometimes y is more manageable for calculations?
Exactly! When you have functions expressed as x = f(y), it's often easier to compute areas vertically. Let's define the area between these curves.
What's the actual formula for this area?
Great question! The formula is A = ∫ from y=c to y=d f(y) dy. Here, c and d are the bounds on the y-axis that you've chosen. Remember this as our new integral approach!
Let’s look at an example. Suppose we want to find the area between the curves defined by x = y² and x = 2y. First, what would we need to determine?
The points where the curves intersect, right? Those give us our limits!
That's correct! We need to solve y² = 2y to find the intersection points. Can someone solve that for us?
The solutions are y = 0 and y = 2. So, we integrate from 0 to 2?
Right again! Now we set up our integral: A = ∫ from 0 to 2 (2y - y²) dy. Can you compute that?
Let’s summarize the steps we’ve taken to find areas based on y. What’s the first step?
Find the points of intersection!
Yes! And then what do we do?
Set up the integral between those limits.
Perfect! Finally, we compute the integral. Remembering to take the upper function minus the lower function is key. Any questions?
Finding the area between the curves x = y² and x = 2y, with limits from y = 0 to y = 2, involves evaluating the integral A = ∫ from 0 to 2 (2y - y²) dy.
To calculate the area between x = sin(y) and x = cos(y) from y = 0 to y = π/2, find the limits and set up the integral.
If y = 3y - y² is defined and intersects at y = 0 and y = 3, the area calculation will involve integrating from these limits.
To find that area, oh can’t you see? Integrate in y, just follow me!
Imagine two rivers flowing; one higher than the other. If you want the land between them, remember to focus on their heights represented vertically!
YIELD: Y-axis means integrating For Limits on y dimensions - this will help calculate area.
FIND: Fun in Notation for Determining areas, ensuring to set limits right.
Term
Area under a curve
Click to flip
Term
Vertical Integration
Click to flip
This section discusses how to calculate the area bound by curves expressed in terms of y, specifically using the integral with respect to y.
In this section, we learn how to find the area between two curves when the function is expressed as x = f(y). The area calculation is performed using definite integrals with limits defined by two values of y.
When given a function as an equation in the form of x = f(y), the process of determining the area between curves changes slightly. Rather than integrating with respect to x, we integrate with respect to y, between the limits defined by the minimum and maximum values of y that bound the area we are interested in.
The area A between the curves from y = c to y = d is calculated as:
A = ∫ from y=c to y=d f(y) dy
This formula is instrumental when analyzing graphs where y is defined explicitly in terms of x, allowing the integration to occur vertically.