Use 6 rectangles to approximate the area under f ( x ) = x 3 f\left(x\right)=x^3 f ( x ) = x 3 from x = 0 x=0 x = 0 to x = 3 x=3 x = 3 with left endpoints.

Let's give this problem a try. So here we're asked to use 6 rectangles to approximate the area under the curve f of x is equal to x cubed from x equals 0 to x equals 3, and we're asked to use left endpoints to make this approximation. Okay. So we're trying to approximate the area underneath this curve here, And the first thing I'm going to do is calculate the base of each rectangle, because I know that the base is going to stay constant in this problem. Now the equation for the base is b minus a over n. Now in this equation here, b is the high point on the interval, so it's the end of the interval right here, which we can see is 3. Now a is the start of the interval, which I can see is 0, and then that's divided by n which is the number of rectangles or 6. So we have 3 minuteus 0 over 6, which turns out to just be 1 half. So all of these rectangles have a base of 1 half. Now at this point, what I can do is start constructing what these rectangles are going to look like on our graph. Now I noticed that each one has a base of 1 half, so that means there's gonna be 6 rectangles going all the way from 0 to 3. So we're going to have halfway marks between each rectangle. So I'm going to divide this, so that way we have 6 points that we're looking at on the x axis. And if I start here, we're here at 0, and then between 0 and 1 is 1 half, between 12 is 3 halves, and between 23 is 5 halves. Now, drawing out these rectangles, we're using left endpoints, so our first rectangle will look something kinda like this. Our next rectangle is going to be right here, and will look something kind of like that. So it's kinda hard to see these first two rectangles. Now our 3rd rectangle is going to go from 1 it's gonna touch at the left endpoint in the or the upper left corner and go all the way over here to 3 halves. Now our 4th rectangle is going to touch the curve of the left endpoint and then go all the way over here to 2. Our 5th rectangle is going to touch the curve right here, and and then we're going to have a left endpoint there and go all the way up to 5 halves. And then our 6th rectangle, the last one is going to touch the curve here and go over to 3. So these are the 6 rectangles we have underneath this curve, and what we need to do is calculate the area of all 6 of these rectangles and then add them up. Now, we can go ahead and start this process by just figuring out what the area is approximately going to be. And again, this will always be an approximation since we're using rectangles to find about the area rather than the true area, or like the exact amount. Now, the area of a rectangle is base times height, and the base is always going to be 1 half. Now as for the height, it's going to be the input of our left endpoints where we are in the x axis plugged into our function f of x is equal to x cubed. Now I can see that we start over here at 0. So our first height is going to be 0 cubed. Then we're going to add this to the next rectangle, which is going to be 1 half times this input, our left endpoint here, which is 1 half cubed. Then we're going to add this to our 3rd rectangle, which has a base of 1 half. All of these rectangles have a base of 1 half. It's gonna be seen right here, and that's going to be the base of 1 half for our 3rd rectangle multiplied by this endpoint, which is going to be one cubed. Then we're going to add this to our 4th rectangle, which has a base of 1 half, and that's going to be this input right here, 3 halves cubed. I'm plugging all of these into this function to get what the height is at each point. Then we're going to add our next rectangle, which has a base of 1 half, and then we have a height of 2 cubed, and then we're going to add this to our next rectangle. So this we have 1, 2, 3, 4, 5 rectangles. So now we're at our last rectangle, which has a base of 1 half, and then we have this input here, 5 halves cubed. So this is going to be all of the rectangles that we have. Notice we've done base times height 6 times for all 6 rectangles. Now, at this point, what I'm going to try and do is simplify what I can, because there's a lot of complicated things and fractions going on here. But starting with this first rectangle here, notice that we have 1 half times 0 cubed. Anything times 0 is just gonna give you 0. So we're going to have 0 plus, and then we're going to have 1 half. And then 1 half cubed, that's the same thing as 1 eighth because 1 cubed is 1 and 2 cubed is 8. Then we're going to have plus 1 half times 1 cube, which is just going to be 1. Then we're going to have plus 1 half times 3 halves cubed. Now 3 cubed, that's going to give you 27. And 2 cubed, we've already discussed, is 8, so that's going to be our next fraction. Then we'll have 1 half plus 2 cubed. 2 cubed is 8, and then we're going to have plus, and then 1 half times 5 cubed, or 5 hats cubed. Then 5 cubed is 125, and then 2 cubed is 8. So So this right here is what happens if we simplify this out a little bit. Now the next thing I'm going to do is I'm going to try to make this even more simple by actually factoring out this 1 half. I think this is going to make these fractions a little bit easier to deal with if I don't have this 1 half in the front each time. So we don't have to write this 0, that's done. But what I can do is factor out this 1 half, and I'm going to write all of these fractions here. So we have 1 eighth, then we have just 1, then we have 27 over 8, then we have this next piece 8, then we have this next piece 125 over 8. So notice how it just looks a little bit more simple when we factor that one half out. I'm just doing this to make the algebra a little bit easier on myself. Now from here, what I'm going to do is try to combine everything inside these parentheses by getting common denominators. So we'll have this one out out in front, then we have 1 eighth, and one is the same thing as 8 over 8, because 8 divided by 8 is 1. So we have 8 over 8 plus 27 over 8. Now 8 times 8 is 64, and then if we divide that by 8, that's the same thing as just 8. So I'm getting 8 in all these denominators, and that's plus 125 over 8. And now everything has a common denominator, so I can just add the numerator straight across. So what we're going to have is we're going to have a 1 half, and then adding these numerators, we'll have 1 +8, which is 9, and then 9 +27, which is 36. We'll have 36 +64, which is a 100, then a 100 plus 125, which is 225, and all of this is going to be over 8. Now my last step for approximate for getting the area approximate under this curve is going to be to multiply this 2 and this 8, which should give you 16. So we'll have 225 over 16, and that right there is the solution. So this is how you can find the area approximate underneath x cubed, from x equals 0 to x equals 3, using left endpoints and 6 rectangles. That's the solution. Hope you found this video helpful, and let's move on.

23. Intro to Derivatives & Area Under the Curve

Area Under a Curve

Precalculus

23. Intro to Derivatives & Area Under the Curve

Area Under a Curve

Struggling with Precalculus?

Join thousands of students who trust us to help them ace their exams!Watch the first video

Multiple Choice

Use 6 rectangles to approximate the area under $f\left(x\right)=x^3$ from $x=0$ to $x=3$ with left endpoints.

$\frac{225}{16}$

$\frac{225}{8}$

$\frac{81}{4}$

Verified step by step guidance

Identify the function f(x) = x^3 and the interval [0, 3] over which you need to approximate the area under the curve using 6 rectangles.

Calculate the width of each rectangle. Since the interval is from x = 0 to x = 3, the total width is 3. Divide this by the number of rectangles, 6, to get a width of 0.5 for each rectangle.

Determine the x-values for the left endpoints of each rectangle. Starting from x = 0, the left endpoints will be x = 0, 0.5, 1, 1.5, 2, and 2.5.

Evaluate the function f(x) = x^3 at each of these left endpoint x-values to find the height of each rectangle. This gives you the heights: f(0), f(0.5), f(1), f(1.5), f(2), and f(2.5).

Calculate the area of each rectangle by multiplying the height by the width (0.5). Sum these areas to approximate the total area under the curve from x = 0 to x = 3.