Approximate the area under the curve f ( x ) = x + 3 f\left(x\right)=\sqrt{x+3} f ( x ) = x + 3 <path d="M95,702c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5,-221l0 -0c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47zM834 80h400000v40h-400000z"> over the interval [ 1 , 5 ] \left\lbrack1,5\right\rbrack [ 1 , 5 ] using the Right Riemann sum with 8 subintervals.

Let's give this problem a try. Here we're asked to approximate the area under the curve of this function over the interval from one to five using a right Riemann sum with eight subintervals. Now in this problem what we have is we have some sort of function that we're dealing with, and we're given what that function is. So if I were to draw a graph right here, notice that our function is f of x is equal to the square root of x plus three. Now in this problem, we're doing a Riemann sum. So we actually don't need a graph if we don't want one, we could just use the equation. But just for reference, I'm gonna go ahead and draw the curve of some function on here. I don't know exactly that it's gonna be this function, but just some random curve, which we'll call f of x is gonna be some function that we're dealing with. We're trying to find the approximate area under the curve of this function from an x value of one all the way up to an x value of five. So this is the area we're trying to estimate right here. Now to do this, what I'm going to do is use a Riemann sum. And the first piece for the Riemann sum is recognizing what the equation is. Now the equation for a right endpoint Riemann sum goes like this is gonna be that r of n is equal to the right endpoint Riemann sum is gonna be the sum from k equals one to n of f of x of k for the right endpoint Riemann sum, and that's gonna be delta x. So in essence, we're adding up a bunch of rectangles, finding their areas is gonna be the height times the width of each individual rectangle. Now what I'm first going to do is knock out this delta x here, because delta x is going to be a constant. The width of each rectangle or subinterval is going to stay the same. Delta x can be calculated like this. It's gonna be b minus a over n. Now b is gonna be the end of our interval here, which is five, minus the start of our interval a, which is one all divided by the number of rectangles, which is the number of subintervals or eight. Now, five minus one over eight which all reduces down to one half. So one half is going to be the result for delta x. So we have a delta x is one half, and I can go ahead and pull this constant outside of the sum. So what that means is that r of n is going to be one half, and then we're going to have the sum from k equals one to n of f of x k. So as you can see, we're starting to get somewhere. We're starting to piece this together. Now what should I do next? Well, the next thing I'm going to do is figure out what this n is, and then I'm going to figure out what my function is going to look like if I take f of x k and replace it with the function we have up here. So doing this, we'll have one half, and then the sum is gonna go from k equals one to n, which is the number of subintervals. So we already said that that's going to be eight. So we'll have from k equals one to eight, and then for f of x k, well, I just need to take x k and plug it in for x in this function to replace it. So we won't have the square root of x of k, and then that's going to be plus three. So that's what we have for the square root. So this right here would be the sum, and this is how you can set things up. So all you would need to do from here is solve this sum. Now the way that we could do this is we need to recognize what's happening here. Notice that we have k equals one, and we're incrementing all the way up to eight. Now what that would mean is we would start with x of one, and then we would have x of two, x of three, x of four, and that would go all the way up to x of eight. But notice that for our subintervals, we're always going up by one half. So if you imagine that we took this curve and we filled it with eight rectangles, and we just filled with a certain number of rectangles, what we're doing is we're starting at the rightmost side of this first rectangle, then we're going over to there, and then we're going over to there. And that's gonna be the strategy that we take for estimating this entire area under the curve. So going back over to here, assuming, again, assuming we didn't have this graph necessarily, we were just trying to use this equation. Well, I'm just gonna go ahead and set things up. So we'll have one half that's gonna be multiplied by this sum as we expand it out. So we're gonna have x of zero first. And x of zero would be where we start on our interval. Now notice we're starting at one, but what I would wanna do is travel one width over since I can see that our width is one half, and we're starting from the right side. So really what we would have is 1.5. So we're going to start with f 1.5, which would be the square root of 1.5 plus three. So that'd be the first piece. Then we would have plus, and I have the square root of two plus three. Then I'd have plus the square root of 2.5 plus three. And this is just gonna keep going, so I'll go down here. So then we're gonna have plus the square root of three plus three. We'll have plus the square root of 3.5 plus three. I'm always going up by point five because that's the same thing as one half. I'm incrementing up by delta x each time I go up one. They're going to have plus. And we're and keep in mind, this is gonna keep adding on here. So we're gonna have plus. And with the square root, so we have 3.5, so we're gonna have four plus three, and then we're gonna have plus the square root of 4.5 plus three. They're going to have plus the square root of five plus three, and this is where the increment is going to stop right here. So notice that these are the numbers that we get. And notice we have one, two, three, four, five, six, seven, eight pieces since we went from one to eight. So we know we did this correctly. So we have these eight square roots that we're dealing with here for these different numbers. What you could do is evaluate everything right here and then multiply it by one half. And if you crunch all of these numbers, you should get approximately 9.96 as your result. So this would be the solution to the Riemann sum using right endpoints. Now the main thing here is to recognize something. If we use these equations for the Riemann sums, for the right endpoints or the left endpoints or even the midpoints, what we can do is we can solve these types of problems where we have to estimate the area underneath the curve of a function without using a graph. We kinda just drew a mock up graph over here for reference, but notice we could have done this completely without the graph. We could have just started with this equation and evaluated the sum and recognized how much we need to increment over by given this delta x right here, and that would have allowed us to solve this problem as well. So that's what's really nice about understanding the Riemann sum notation and how to solve these problems. So hope you found this video helpful. 9.96 is the approximate area of solution for this problem, and let's go ahead and move on and get some more practice. See you in the next one.

8. Definite Integrals

Riemann Sums

Business Calculus

8. Definite Integrals

Riemann Sums

Struggling with Business Calculus?

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Multiple Choice

Approximate the area under the curve $f\left(x\right)=\sqrt{x+3}$ over the interval $\left\lbrack1,5\right\rbrack$ using the Right Riemann sum with 8 subintervals.

9.75

9.96

9.54

9.72

Verified step by step guidance

Divide the interval [1, 5] into 8 equal subintervals. To do this, calculate the width of each subinterval, Δx, using the formula Δx = (b - a) / n, where a = 1, b = 5, and n = 8.

Determine the right endpoints of each subinterval. These are the x-values at which the function will be evaluated. The right endpoints are calculated as x_i = a + i * Δx for i = 1, 2, ..., n.

Evaluate the function f(x) = √(x + 3) at each of the right endpoints. This means substituting each x_i into the function to find f(x_i).

Multiply each function value f(x_i) by the width of the subinterval, Δx. This gives the area of each rectangle in the Right Riemann sum approximation.

Add up all the areas of the rectangles to approximate the total area under the curve. This sum is the Right Riemann sum approximation for the given function over the interval [1, 5].