7. Applications of Integrals

Let's start with the area under a curve. Can anyone tell me what we mean by the 'area under the curve'?

Is it like finding the total space between the curve and the x-axis?

Exactly! If we have a function 𝑓(𝑥) that is continuous over an interval [𝑎, 𝑏], the area can be calculated using the definite integral: $Area = \int_{a}^{b} f(x) \, dx$. This gives us the 'signed area'—meaning if the curve is below the x-axis, we get a negative value.

What if part of the curve is below the axis?

Good question! We take the absolute value of the integral when calculating area, so the area is always positive. Remember: 'UP-ABSOLUTES!' helps us keep this in mind!

Now that we know how to find the area under a single curve, let’s explore how to find the area between two curves. What do we need to do to calculate that?

I think we need to know the equations of both curves, right?

Correct! If we have two curves, 𝑓(𝑥) and 𝑔(𝑥), where 𝑓(𝑥) is above 𝑔(𝑥), the area between them is given by: $Area = \int_{a}^{b} [f(x) - g(x)] \, dx$. It’s important to first identify the points of intersection to determine our limits, [𝑎, 𝑏].

So we sketch the curves first to see where they intersect?

Yes! Visualizing is crucial, as it helps determine how the functions relate. Remember, 'Sketch, Find, Solve!' to keep this process in mind.

Sometimes, we find areas that are bounded by axes as well as curves. Can anyone explain how we might calculate that?

Do you just use the same integral method as before?

Almost! Instead, we use the formula: $Area = \int_{a}^{b} |f(x)| \, dx$. If the function changes sign between 𝑎 and 𝑏, we’ll split the integral at those points where the function crosses the axis. 'Always Positive!' helps remind us of this.

And for functions expressed in the y-direction?

Great point! If we are given 𝑥 = 𝑓(𝑦)$ and need to find area between two y-values, we would use the integral with respect to y: $Area = \int_{c}^{d} f(y) \, dy$. That's a different perspective on the same concept!