Practice Exercise

Today we're going to delve into how to find areas under curves using integrals. Can anyone tell me what the formula for calculating the area under a curve is?

Isn't it the integral of the function from a to b?

Exactly, it's represented as $\int_a^b f(x) \, dx$. Remember, this gives us the signed area. What's important to note about signed areas?

If the curve is above the x-axis, the area is positive, and if it’s below, it’s negative!

Correct! And to find actual area, we'd take the absolute value when needed. Let's move on to our first exercise.

Q1: Find the area bounded by the curve $y = \sqrt{x}$, the x-axis, and the lines $x = 1$ and $x = 4$. Who would like to start?

Now let's discuss how to calculate the area between two curves. Can someone remind me how we set this up?

We use the formula $\int_a^b [f(x) - g(x)] \, dx$ where $f(x)$ is the upper function and $g(x)$ is the lower one.

Great! And why is sketching the curves first helpful?

It helps us see where they intersect and identify which function is on top.

Exactly. Let's practice with Q2: Find the area between the curves $y = \sin(x)$ and $y = \cos(x)$ from $x = 0$ to $x = \frac{\pi}{2}$.

Next, we'll tackle bound areas like those enclosed by a parabola and a line. Why is understanding where the functions intersect crucial?

Because those points give us the limits of integration!

Absolutely! For example, in Q3, we need to find the area enclosed by the parabola $y = x^2$ and the line $y = 4$. What will we do first?

We find the points where they intersect!

Correct! Let's work through it.

Finally, let’s discuss cases where we have absolute value functions, such as $y = |x|$. What's important when calculating areas here?

We need to take the absolute value because the function changes sign!

Exactly! For Q4, we need the area bounded by $y = |x|$ and the lines $x = -2$ and $x = 2$. How will we approach this?

We’ll split the integral into two parts: one for $x < 0$ and one for $x > 0$!

Well done! Let’s work through the integrals together.

To wrap up, let’s do one more example together—Q5: Calculate the area between $y = e^x$ and $y = e^{-x}$ from $x = -1$ to $x = 1$. What should we do first?

We should determine which function is on top over that interval, right?

Exactly! After that, we can set up the integral. Who wants to take the lead on this question?

I can! Let's see the setup.

Great! Let's do it together and then review all key concepts before our quiz next week.