yego.me
💡 Stop wasting time. Read Youtube instead of watch. Download Chrome Extension

Finding definite integrals using area formulas | AP Calculus AB | Khan Academy


3m read
·Nov 11, 2024

  • [Instructor] We're told to find the following integrals, and we're given the graph of f right over here.

So this first one is the definite integral from negative six to negative two of f of x dx.

Pause this video and see if you can figure this one out from this graph.

All right, we're going from x equals negative six to x equals negative two, and the definite integral is going to be the area below our graph and above the x-axis.

So it's going to be this area right over here. And how do we figure that out?

Well, this is a semicircle, and we know how to find the area of a circle if we know its radius.

And this circle has radius two, has a radius of two.

No matter what direction we go in from the center, it has a radius of two.

And so the area of a circle is pi r squared.

So it would be pi times our radius, which is two squared, but this is a semicircle, so I'm gonna divide by two.

It's only 1/2 the area of the full circle.

So this is going to be four pi over two, which is equal to two pi.

All right, let's do another one.

So here we have the definite integral from negative two to one of f of x dx.

Pause the video and see if you can figure that out.

All right, let's do it together.

So we're going from negative two to one, and so we have to be a little bit careful here.

So the definite integral, you could view it as the area below the function and above the x-axis.

But here the function is below the x-axis.

And so what we can do is, we can figure out this area, just knowing what we know about geometry, and then we have to realize that this is going to be a negative value for the definite integral because our function is below the x-axis.

So what's the area here?

Well, there's a couple of ways to think about it.

We could split it up into a few shapes.

So you could just view it as a trapezoid or you can just split it up into a rectangle and two triangles.

So if you split it up like this, this triangle right over here has an area of one times two times 1/2.

So this has an area of one.

This rectangle right over here has an area of two times one, so it has an area of two.

And then this triangle right over here is the same area as the first one.

It's going to have a base of one, a height of two, so it's one times two times 1/2.

Remember the area of a triangle is 1/2 base times height.

So it's one.

So if you add up those areas, one plus two plus one is four, and so you might be tempted to say oh, is this going to be equal to four?

But remember our function is below the x-axis here, and so this is going to be a negative four.

All right, let's do another one.

So now we're gonna go from one to four of f of x dx.

So pause the video and see if you can figure that out.

So we're gonna go from here to here, and so it's gonna be this area right over there.

So how do we figure that out?

Well, it's just the formula for the area of a triangle, base times height times 1/2.

So or you could say 1/2 times our base, which is a length of, see we have a base of three right over here, go from one to four, so 1/2 times three times our height, which is one, two, three, four, times four.

Well, this is just going to get us six.

All right, last but not least, if we are going from four to six of f of x dx.

So that's going to be this area right over here, but we have to be careful.

Our function is below the x-axis, so we'll figure out this area and then it's going to be negative.

So this is a half of a circle of radius one.

And so the area of a circle is pi times r squared, so it's pi times one squared.

That would be the area if we went all the way around like that, but this is only half of the circle, so divided by two.

And since this area is above the function and below the x-axis, it's going to be negative.

So this is going to be equal to negative pi over two.

And we are done.

More Articles

View All
Perfect Your Desires
One of the things I’ve learned relatively recently in life is that it’s way more important to perfect your desires if you want to do something than it is to try to do that thing when your desire is not 100%. An example would be like… you know, self-disci…
How To Make Every Day Count
Are you living your best life, or are you waiting for it to happen? How we spend our life is, in fact, how we spend our days. But many people find that out too late. They sacrifice their whole life and get nothing in return. So many people spend their liv…
The Stoic Truth: Are You Sabotaging Your Own Success? | STOICISM INSIGHTS #stoicism
Welcome back to Stoicism Insights, your guide to unlocking the timeless wisdom of Stoic philosophy for a more fulfilling life. Today we’re delving into a topic that’s often overlooked: the negative habits that hinder our journey towards virtue and tranqui…
Humanity's Fascination with Mars | MARS
Dreamers of space have always had their eyes there, their hopes, their aspirations on getting to Mars. It has to look at the sky, saw that thought, and wondered what’s on it. As soon as people understood what planets were, some of them said, “Wouldn’t it …
Elephant Cleverly Steals Sugar Cane off a Truck in Thailand | Secrets of the Elephants
Thailand Highway 3259 is a sugarcane transport road. Thousands of farmers use it to get their crops to the refinery. But this highway has a toll collector. Locals call him the Don. And this is his territory. He’s a master dealmaker, calculating risk vers…
Why “Looking Poor” Is Important
What’s up you guys, it’s Graham here. In the last few months, you might have come across one of these videos: the importance of looking poor. After all, when you really dig into it, it is insane how many people these days are pretending to be rich, diggi…