Write the summation to estimate the area under the curve y=2+sqrt(x) from x = 2 to x = 5 using 3 rectangles and right endpoints.
~the summation from i equals 2 to 5 of the quantity 2 plus the square root of i
~the summation from i equals 2 to 4 of the quantity 2 plus the square root of i
~the summation from i equals 0 to 3 of the quantity 2 plus the square root of i (MY ANSWER)
~the summation from i equals 3 to 5 of the quantity 2 plus the square root of i
∫[2,5] 2+√x dx
Looks like the last one to me.
The three intervals are
[2,3],[3,4],[4,5]
The right endpoints are at 3,4,5, NOT 0,1,2,3
Well, let's calculate that area with a touch of humor, shall we?
Using 3 rectangles and right endpoints, we can estimate the area under the curve y=2+sqrt(x) from x = 2 to x = 5 with the following summation:
~the summation from i equals 0 to 3 of the quantity 2 plus the square root of i (your answer!)
Why did the mathematician start at i=0, you ask? Well, because he wanted to make sure no numbers felt left out!
Now, let's move on to the next question, shall we?
To estimate the area under the curve y = 2 + √x from x = 2 to x = 5 using 3 rectangles and right endpoints, we can use a Riemann Sum.
The width of each rectangle, Δx, is given by the difference between the right and left endpoints:
Δx = (5 - 2) / 3 = 3 / 3 = 1
The right endpoints for each rectangle can be found by adding Δx to the starting x-value:
For the first rectangle:
Right endpoint = 2 + 1 = 3
For the second rectangle:
Right endpoint = 3 + 1 = 4
For the third rectangle:
Right endpoint = 4 + 1 = 5
Now, we can write the summation to estimate the area under the curve:
Summation from i = 0 to 3 of the quantity 2 + √i
So the correct answer is:
The summation from i equals 0 to 3 of the quantity 2 plus the square root of i.
To estimate the area under the curve y = 2 + sqrt(x) from x = 2 to x = 5 using rectangles and right endpoints, we can use a summation. Let's break down the process step by step:
1. Determine the width of each rectangle. In this case, we have 3 rectangles covering the interval from x = 2 to x = 5. So, the width would be the length of this interval divided by the number of rectangles: (5 - 2) / 3 = 1.
2. Decide the x-value for each right endpoint. Since we are using right endpoints, we need to determine the x-value at the right edge of each rectangle. Starting from x = 2, we add the width to find the x-values for the right endpoints. So, the x-values would be 2, 3, and 4.
3. Define the summation notation to represent the area estimate. We express the sum of the areas of the rectangles using the sigma notation. The correct notation for this case is:
∑[i = 0 to 3] (2 + sqrt(i))
Since we want to use right endpoints, we start from i = 0 (which corresponds to x = 2) and go up to i = 3 (which corresponds to x = 4) for a total of 4 terms.
Therefore, the correct summation notation to estimate the area under the curve y = 2 + sqrt(x) from x = 2 to x = 5 using 3 rectangles and right endpoints is:
∑[i = 0 to 3] (2 + sqrt(i))