Use 3 rectangles to approximate the area under f ( x ) = x 2 f\left(x\right)=x^2 from x = 2 x=2 to x = 5 x=5 with right endpoints.

Alright. So in this problem, we're asked to use 3 rectangles to approximate the area underneath this function, f of x equals x squared from x equals 2 to x equals 5. So let's see how we can go about doing this. Now typically, the first step that I like to do when solving these types of problems is to figure out what the base of my rectangle is going to be. And the reason for this is because if I can calculate the base, that is going to be the same for every single rectangle that we have. And recall that the base is going to be b minus a over n. Now b, that's going to be at the end of the interval, which I can see is over here at 5. A is going to be the start of the interval, which is at 2, and then n is the number of rectangles, which I can see there is 3 rectangles. So that's a, that's b, and then that's n. Now putting all of this in, we get 5 minus 2 over 3, which comes out to just 1. So the base of each rectangle is going to be 1. Now what I'm going to do from here is I'm going to fill this this curve that we have with rectangles from 2 to 5 on the x axis. So we'll start over here at 2, and notice that we're asked to do a right endpoint approximation. So I'm going to use right endpoints, meaning, we'll start over here, we have we're starting at 2, but I need to travel from here to 3 because this would be a rectangle at the right endpoint. So we go here at 3, we'd be right there, and then this right here would be a right endpoint approximation for this rectangle. Now I'm going all the way up to 5. Each of these have a base of 1, so we'll do another rectangle right about there, and then we'll do another rectangle right here. So notice how we have rectangles that start at x equals 2 and go over to x equals 5 from right endpoints touching the curve. Now what we need to do is figure out what this area is going to be for each of these rectangles that we have, and that's what we're trying to calculate to get this approximation. So recall that the area of a rectangle is base times height. So if I want to find the total area for this entire curve that we have, well, the area is going to be approximately equal to base times height of each rectangle. So we're going to have a base of 1. We can see that's going to be the base for every rectangle we have times this height. Now this height right here is going to be whatever this value is squared, because our function is f of x squared. So we're going to have 3 squared, which is 9. So we're going to have 1 times 9 for our first rectangle, then we're going to add this to our base of 1 times the height of the second rectangle, which I can see that's going to be 4 squared, which is 16. Next, I'm going to add this to our the area of our 3rd rectangle, which is going to be the base of 1. All of these rectangles have the same base, and that's going to be multiplied by this height, which is going to be 5 squared or 25. So this right here is going to be the approximate area underneath this curve using 3 rectangles with right endpoints. Now 1 times 9, this is gonna be 9, so we'll have 9 plus 16, which is 25, and then that's gonna be 25 plus 25, which is approximately 50. Now 50 is actually the answer here, but the reason it's approximately is because we're approximating the area under this curve. So this is the true calculation if we're using rectangles to approximate, and that is how you can solve this problem. So hope you found this video helpful, and let's move on.

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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

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Multiple Choice

Use 3 rectangles to approximate the area under $f (x) = x^{2}$ from $x equals 2$ to $x equals 5$ with right endpoints.

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Verified step by step guidance

Identify the interval [2, 5] and divide it into 3 equal subintervals. The width of each rectangle (Δx) is calculated as (5 - 2) / 3.

Calculate the right endpoints of each subinterval. For the first subinterval, the right endpoint is x = 3. For the second, x = 4. For the third, x = 5.

Evaluate the function f(x) = x^2 at each right endpoint to find the height of each rectangle. Calculate f(3), f(4), and f(5).

Multiply the height of each rectangle by the width (Δx) to find the area of each rectangle. This gives the areas: f(3) * Δx, f(4) * Δx, and f(5) * Δx.

Sum the areas of the three rectangles to approximate the total area under the curve from x = 2 to x = 5.