Use five rectangles to estimate the area under the curve of f ( x ) = x 2 f\left(x\right)=x^2 f ( x ) = x 2 from x = 0 x=0 x = 0 to x = 5 x=5 x = 5 using left endpoints.

So in this problem, we're asked to use five rectangles to estimate the area under this function curve from x equals zero to x equals five using the left endpoints. Okay. Now what I'm gonna start by doing is first figuring out what the width of my rectangles are going to be that go underneath the curve of this function. And recall, this is gonna be the high value minus the low value divided by the number of rectangles that we have. Now I can see that my high value goes all the way up to five. I can see my low value goes to zero, and then I can see the number of rectangles that we're dealing with according to this problem is five. So we have five minus zero over five, which is one. So every rectangle we deal with in this problem will have a length of one. Now going over from here, this is where we start the interval at zero, and then we go all the way up to five, and we need to fill this with five rectangles. So this rectangle is going to have a width of one, and we'll touch at the left side of that rectangle. This rectangle will have a width of one, and we touch this left side. This rectangle will also have a width of one, and we can just go ahead and keep going like this. So we're gonna have this rectangle right about here that touches the endpoint there. And then our last rectangle is gonna go all the way over to this side, and that's where we're going to finish. So this right here would be how we could fill the area, how we could fill our triangles here with rectangles. And what we're trying to do is find the area of each one of these rectangles. Then what we're gonna do is find once we find the areas, is we're gonna add everything up to get the approximate area underneath the curve of this function. Now to find this, well, we're gonna do it like this. So we need to find the approximate area underneath the curve of the function. And so what I need to do is first find f of zero, and that's gonna be this value right here. And then I need to multiply that by delta x, the width of my rectangle. And I'm actually gonna do it like this. I'm gonna actually take this delta x and factor it out because I know that delta x is gonna be multiplied by every function value that we have because delta x is the same value. So we have delta x times this height here. So we have width times height, and it's going to be plus, and then we're going to have this left value for this rectangle. This is gonna be plus f of one. We're gonna have plus f of two, this next value right here. We're gonna have plus f of three. I'm just going up to each rectangle. Then we're gonna have plus f of four, and that's gonna be where we stop right about there because notice that this is the leftmost side of this rectangle, so these are going to be the values that we plug in. Now delta x value, we already figured out that that's a value of one. This was something that we calculated earlier. But finding these function values, well, we need to take every input that we have and plug it into our function, which is x squared. So to do this, we're first gonna have zero squared, which is zero. Then we're going to have one squared, which is one. Then we're going to have two squared, which is four. They're going to have three squared, which is nine. And then four squared, which is 16. And you could figure that out by also looking at these values here. This would be at one. This would be at four. This would be at nine, and so on. Right? You can kinda see how the graph where it's gonna end up, but this is a way that we can just plug it into our function and get the true values. So let's go ahead and do the math here. So we have one on the outside. Here we're gonna have zero plus one plus four, and we're gonna have, within one plus four is five, and then plus nine, that's 14. So we're gonna have 14 plus 16, which turns out to be 30. So we're gonna have one times 30, which is 30, and then that's going to be the solution to our problem right there. So the area underneath our curve is going to be approximately 30 square units. So this should actually be an approximate sign right here that we're using because this isn't a truly accurate answer, but this is an approximation for what we're getting. Right? The true area is not has not been calculated, but what we do with this this rectangle approximation is find about how much area that we have. Notice there's a little area we didn't account for because that's what using rectangles does. So it gives us an approximate answer for the area underneath this curve, and that is the solution to the problem. So this is how you can do this process where you approximate the area under a curve by using rectangles. Hope you found this video helpful, and let's get some more practice.

8. Definite Integrals

Estimating Area with Finite Sums

Business Calculus

8. Definite Integrals

Estimating Area with Finite Sums

Struggling with Business Calculus?

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Multiple Choice

Use five rectangles to estimate the area under the curve of $f\left(x\right)=x^2$ from $x=0$ to $x=5$ using left endpoints.

$A=30$

$A=55$

$A=41.25$

$A=41.67$

Verified step by step guidance

Step 1: Understand the problem. You are tasked with estimating the area under the curve f(x) = x^2 from x = 0 to x = 5 using five rectangles and the left endpoint method. The graph of f(x) = x^2 is provided.

Step 2: Divide the interval [0, 5] into 5 equal subintervals. The width of each rectangle (Δx) is calculated as Δx = (5 - 0)/5 = 1.

Step 3: Determine the left endpoints of each subinterval. The left endpoints are x = 0, x = 1, x = 2, x = 3, and x = 4.

Step 4: Evaluate the function f(x) = x^2 at each left endpoint to find the heights of the rectangles. The heights are f(0) = 0^2, f(1) = 1^2, f(2) = 2^2, f(3) = 3^2, and f(4) = 4^2.

Step 5: Multiply the height of each rectangle by the width (Δx = 1) and sum the areas of all rectangles to estimate the total area under the curve. The estimated area is Σ [f(x_i) * Δx] for i = 0 to 4.