Use three rectangles to estimate the area under the curve of f ( x ) = 1 3 x 3 f\left(x\right)=\frac13x^3 from x = 0 x=0 to x = 3 x=3 using the right endpoints.

this problem, we're asked to use three rectangles to estimate the area under the curve of this function, f of x equals one third x cubed, and we're asked to do this from x equals zero to x equals three using right endpoints. So let's see how we can figure this out. Now what I'm trying to do is estimate the approximate area under this curve, and we're asked to use three rectangles to do it. Each rectangle is going to have a width delta x. And the way that I can find this delta x, this width, is going to be the high point, which is three, minus the low point, which is zero, all divided by the number of rectangles, which is three. So I have three minus zero over three, which is one. Right? And this is how you can always find the width of these rectangles when you're using these rectangle approximations for the area under a curve. So we know that delta x is going to be one. We also know that we're using right endpoints. So what I can do is go over to this function here, and I can go ahead and start here from zero, and I'm going all the way up to three, which is going to actually be somewhere past what the graph is that we can see. So it's gonna be somewhere about there. So what I can do is fill this with rectangles. Now, I'll start by filling this over here at zero, and we're going to go all the way up to one. So our curve's gonna look something like, or a rectangle's gonna look something like this. Now notice how a rectangle goes over the curve on this left side here. That's because we're using right endpoints to approximate rather than left. So because we're using right endpoints, it's always gonna be touching this right side. So we'll now go over another one unit, which is gonna be right about there, and this is gonna be our second rectangle. And then we'll have our third rectangle, which we go over one unit now, It's gonna go up and it's gonna touch our curve, and then it's gonna go right about here. So this is how we could fill our curve with rectangles using these right endpoints. So now we fill this up with rectangles, and we need to approximate what the area is under this curve by finding the area of each one of these rectangles and adding them all up. Now we can do this using this process that we continue to use, where we just find the area of each one individually. So our approximate area underneath the curve of this function is going to be Area one plus area two, plus area three, since we have three rectangles. Now the area of each rectangle is width times height. The width is always gonna be Delta X for each rectangle, they all have the same width. Now, as for the heights, well, that's going to be whatever the function values are at the right endpoints. So, first off, we're going to have f of one, which is this rightmost endpoint for the rectangle. Then we're going to have plus f of two, which is this rightmost endpoint. Then we're going to have plus f of three, which is that rightmost endpoint. So this is going to be the function values that we or the the inputs that we have for each function, and we need to find the function values to get the height. Notice it's width times height for each rectangle that we have. Now solving this, we know that delta x is one. We already calculated that. And f of one, well, we're just gonna take one and plug it into this function, one third x cubed. So I have one third times one cubed, which is all just gonna come out to one third or about point three three. Then we're going to have plus f of two, which is going to be one third of two cubed. And that will all come out to 2.67 approximately. And just rounding these to decimals, then we're gonna have plus f of three, which plug in three into this function, you should get nine as your result. So we have nine plus 2.67 versus point three three, all multiplied by one. And this is all going to come out to 12 as your result. So 12 is the approximate area under the curve of this function, using right endpoints and three rectangles, each with a width of one. So that is how you can estimate this area. Hope you found this video helpful.

8. Definite Integrals

Estimating Area with Finite Sums

Calculus

8. Definite Integrals

Estimating Area with Finite Sums

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Multiple Choice

Use three rectangles to estimate the area under the curve of $f (x) = \frac{1}{3} x^{3}$ from $x equals 0$ to $x equals 3$ using the right endpoints.

$A equals 3$

$A equals 12$

$A equals 6.5$

$A equals 5.3$

Verified step by step guidance

First, identify the function given: $ f(x) = \frac{1}{3}x^3 $. We need to estimate the area under this curve from $ x = 0 $ to $ x = 3 $ using three rectangles and the right endpoints.

Divide the interval $[0, 3]$ into three equal subintervals. Each subinterval will have a width of $ \Delta x = \frac{3 - 0}{3} = 1 $.

Determine the right endpoints of each subinterval. For the subintervals $[0, 1]$, $[1, 2]$, and $[2, 3]$, the right endpoints are $ x = 1 $, $ x = 2 $, and $ x = 3 $ respectively.

Calculate the height of each rectangle using the function $ f(x) $ at the right endpoints: $ f(1) = \frac{1}{3}(1)^3 $, $ f(2) = \frac{1}{3}(2)^3 $, and $ f(3) = \frac{1}{3}(3)^3 $.

The area of each rectangle is given by $ \text{height} \times \Delta x $. Sum the areas of the three rectangles to estimate the total area under the curve: $ A = f(1) \times 1 + f(2) \times 1 + f(3) \times 1 $.