Use two rectangles to estimate the area under the curve of f ( x ) = 1 2 x 2 f\left(x\right)=\frac12x^2 from x = 0 x=0 to x = 3 x=3 using left endpoints.

So in this problem, we're asked to use two rectangles to estimate the area under the curve of this function right here from x equals zero to x equals three using left endpoints. Now, what we can do is go over to our graph here, and we can take a look at what this function looks like. Now, what I'm trying to do is I'm trying to approximate the area underneath the curve of this function, from an x value of zero, all the way up to three. So this is what I'm trying to find. Now, the way that I'm going to do this is by filling it with rectangles. Now, the question is, is we need to figure out what the dimensions of these rectangles are, what the widths and the heights are going to be, and that's how we can estimate the area. Well, I'm gonna start by figuring out what the width of each rectangle is going to be. I'm going to use this equation, b minus a over n, the number of rectangles we have. Now in this problem, I can see that b, which is the high point that we reach, is going to be three. I can see a, which is the low point, is going to be zero, and I'm gonna divide that by the number of rectangles. And the number of rectangles that I can see is two rectangles. That's how many rectangles we're asked to use, so it's gonna be divided by two. So we have three minus zero over two, which is gonna be three halves or 1.5. So 1.5 is gonna be the width of every rectangle that we're dealing with in this problem. And since we're trying to fill this with two rectangles using a left endpoints, well, let's go over to our graph and do that. So what we're going to have is a rectangle that fills the curve, something like this. We're gonna start from the left end point of this rectangle, and this is gonna go all the way over here to this value of 1.5 on the x axis. Notice it's between one and two. And then we're going to have another rectangle which goes ahead and touches this curve, and it goes all the way to the number three on this x axis. So this is a width of 1.5 as well to get all the way to three. So we go to one and a half and all the way over to three, and then that's going to be our rectangle. So we're trying to estimate this area right here. So let's go ahead and do this. Now the area is going to be approximately equal to the area of these two rectangles, so we're gonna have area one plus area two. And we know for each area, it's width times height. So we have width one times height one, plus width two times height two. Now the width is just going to be this delta x that we calculated, and the width for each rectangle is the same. So I can factor that w one and w two out as just delta x. And then all I'm gonna do is multiply this by height one plus height two. Now each of these heights corresponds to the function value on the left endpoints. So you can see the left endpoints is zero, so we're going to have f of zero. We're going to have plus this endpoint, which is 1.5. So f of zero plus f of 1.5. Now notice that this is in essence what we're finding the area of since we have the width times height one plus the width times height two. So let's go ahead and figure out what these numbers are. Now delta x, we already figured out is 1.5. So 1.5 is gonna go right here. Now as for f of zero plus f of 1.5, well, we can find those by plugging in whatever this number is into our function up here. And you see that our function is one half x squared. So we're going to have one half of zero squared, which is just going to be zero. They're going to have plus, and then we're going to have one half of 1.5 squared. And 1.5 squared times one half, that's all going to come out to 1.125. Now you can crunch these numbers right here, add these together, multiply it by 1.5, and you should get that your final result is equal to 1.6875. So this is the approximate area underneath the curve of the function f of x. Now again, this is just an approximation because as you can see, there's a bunch of area here we did not actually account for when we did this area approximation. We only found the area of, well, basically this rectangle, since this rectangle is actually touching the x axis and it came out to zero for area. But that's the process of using these left endpoints and approximating the area is we're not gonna get a truly accurate answer, but we're just estimating using these rectangles. And we didn't use a lot of rectangles, we only used two, So we didn't get a super accurate estimation. But if you were asked to estimate with two with two rectangles using left endpoints, this would be how you could do it. So hope you found this video helpful, and let's try getting some more practice.

8. Definite Integrals

Estimating Area with Finite Sums

Calculus

8. Definite Integrals

Estimating Area with Finite Sums

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Multiple Choice

Use two rectangles to estimate the area under the curve of $f (x) = \frac{1}{2} x^{2}$ from $x equals 0$ to $x equals 3$ using left endpoints.

$A equals 1.6875$

$A equals 4.21875$

$A equals 8.4375$

$A equals 4.5$

Verified step by step guidance

First, identify the function given: $ f(x) = \frac{1}{2}x^2 $. We need to estimate the area under this curve from $ x = 0 $ to $ x = 3 $ using two rectangles and left endpoints.

Divide the interval $[0, 3]$ into two equal subintervals. Each subinterval will have a width of $ \Delta x = \frac{3 - 0}{2} = 1.5 $.

For the first rectangle, use the left endpoint of the first subinterval, which is $ x = 0 $. Calculate the height of the rectangle using $ f(0) = \frac{1}{2}(0)^2 = 0 $. The area of the first rectangle is $ \text{height} \times \text{width} = 0 \times 1.5 = 0 $.

For the second rectangle, use the left endpoint of the second subinterval, which is $ x = 1.5 $. Calculate the height of the rectangle using $ f(1.5) = \frac{1}{2}(1.5)^2 = \frac{1}{2} \times 2.25 = 1.125 $. The area of the second rectangle is $ 1.125 \times 1.5 = 1.6875 $.

Add the areas of the two rectangles to estimate the total area under the curve: $ 0 + 1.6875 = 1.6875 $. This is the estimated area using two rectangles with left endpoints.