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

this problem, we're asked to use four rectangles to estimate the area under the curve of our function f of x equals x squared plus two, from x equals zero to x equals two using left endpoints. So we're trying to approximate the area under the curve of this function, and the way we can do that is by filling this function with rectangles. Now, the rectangles are each going to have a width of delta x, and the delta x can be calculated by this, b minus a over n, this is the equation. Now b is the end of our interval, which I can see is two. A is the start of our interval, which is zero, and then we divide that by the number of rectangles we'll have in this problem, which is four. So we have two minus zero over four, which is gonna be one half or 0.5. So each rectangle that we deal with has a width of 0.5. So going over to our graph, we're going to start at this leftmost point here because we're going from zero to two. So we're gonna start at this leftmost point, this is going to touch our graph, and notice we're using these left endpoints. So we start here at zero, and then what we're gonna do is go over here to point five, and so we're going to touch on the left side here, go back down. Then we're going to have another rectangle right here at one that goes right here, that touches our graph, goes to the side, it goes back down. And then we'll go over to 1.5, and this is gonna be our last rectangle right here, that's going to fill our function since we have four rectangles. So we now need to figure out what the area is for each one of these rectangles. And if we add all these rectangles up, this will give us the approximation for the area underneath the curve of this function f of x. So let's go ahead and do that. Now, I know that the area, the approximate area underneath the curve of this function is going to be area one, plus area two, plus area three, plus area four, the area of each individual rectangle. We know each rectangle is going to have an area of width times height. Now rather than just writing width one times height one plus width two times height two, I think we all know how to do that. Notice all of these rectangles have the same width. Each of these rectangles have a width of 0.5, and that's gonna be our delta x. So I'm just gonna take delta x, I'm gonna automatically factor it out to the front of this equation. Now, what exactly are the heights going to be? So it's width times height. Well the height of each rectangle is going to be the function output associated with the left side of each rectangle. So what we're going to have is f of zero at the start here. Then we're gonna have plus f of 0.5, which is gonna be this piece right here. And keep in mind, we're using left endpoints. Then we're going to have f of one, plus f of one. That's going to be this one right here. Then we're going to have plus f of this last left end point 1.5, all within our function, and then that's going to be the heights of each rectangle. So notice we're constantly doing width times height, plus width times height, plus width times height, and so on. Now delta x delta x is 0.5. We already calculated this. So we're gonna have 0.5, and then for each of these function values, well, I need to plug these into the function that we have to get what the output is. So we're going to have first f of zero and zero squared plus two. Well, that's just going to give us two. Then we're going to have 0.5 squared plus two, zero point five squared plus two, that's going to give you 2.25. Then we're going to have one squared plus two, which is going to give us three. Then we're going to have 1.5 squared plus two, which is going to give us 4.25. All I'm doing is taking each of these numbers that I see within these functions and plugging it into our function f of x up here. So that's how I got those outputs. Now, from here, these are numbers that we can crunch. We can add all these up, and we can multiply these by 0.5. Feel free to check this on your calculator to see the result that you get. And you should get 5.75 as the approximate area under the curve of this function. So this is the solution to the problem, and that's how we estimated the area under the curve of this function right here. Now notice we do have some extra area underneath the curve of this function. Again, this is not a completely accurate result, but this allows us to approximate about what the area is. So that is the process for solving these types of problems. 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 four rectangles to estimate the area under the curve of $f (x) = x^{2} + 2$ from $x equals 0$ to $x equals 2$ using left endpoints.

$A equals 5.00$

$A equals 6.67$

$A equals 5.75$

$A equals 7.75$

Verified step by step guidance

Identify the function f(x) = x^2 + 2 and the interval [0, 2] over which we need to estimate the area under the curve using four rectangles.

Divide the interval [0, 2] into four equal subintervals. Each subinterval will have a width of Δx = (2 - 0) / 4 = 0.5.

Determine the left endpoints of each subinterval. These will be x_0 = 0, x_1 = 0.5, x_2 = 1, and x_3 = 1.5.

Calculate the height of each rectangle using the function value at the left endpoint of each subinterval: f(x_0), f(x_1), f(x_2), and f(x_3).

Compute the area of each rectangle as the product of the width (Δx) and the height (f(x) at the left endpoint), then sum these areas to estimate the total area under the curve.