Approximate the area under the curve y=2/x from 1 to 2 using ten approximating rectangles of equal widths and right endpoints. Round your answer to the nearest hundredth
delta x = .1
x range ____ y at right
1.0 to 1.1 ____ 2/1.1 = 1.82 times .1 = .182
1.1 to 1.2 ____ 2/1.2 = 1.67 times .1 = .167
1.2 to 1.3 etc
in the end add them up .182 + .167 etc
To approximate the area under the curve \(y=\frac{2}{x}\) from 1 to 2 using ten approximating rectangles of equal width and right endpoints, we can use the Right Endpoint Rule for approximating definite integrals.
1. First, we need to calculate the width of each rectangle. The interval from 1 to 2 is divided into 10 parts, so the width of each rectangle is \(\frac{{2-1}}{10} = \frac{1}{10}\).
2. Next, we evaluate the function \(y=\frac{2}{x}\) at the right endpoints of each rectangle to calculate the height of the rectangles.
- At the first interval, the right endpoint is \(x=1 + \frac{1}{10} = 1.1\). So the height of the first rectangle is \(y=\frac{2}{1.1}\).
- At the second interval, the right endpoint is \(x=1 + \frac{2}{10} = 1.2\). So the height of the second rectangle is \(y=\frac{2}{1.2}\).
- Continue this process until the tenth interval where the right endpoint is \(x=1 + \frac{10}{10} = 2\).
3. Now, we can calculate the area of each rectangle by multiplying the width by the height for each rectangle.
- For the first rectangle, the area is \( \left( \frac{1}{10} \right) \left( \frac{2}{1.1} \right) \).
- For the second rectangle, the area is \( \left( \frac{1}{10} \right) \left( \frac{2}{1.2} \right) \).
- Continue this process until the tenth rectangle where the area is \( \left( \frac{1}{10} \right) \left( \frac{2}{2} \right) \).
4. Finally, we sum up the areas of all 10 rectangles to get the approximate area under the curve.
- Add up the areas of all 10 rectangles: \(\left( \frac{1}{10} \right) \left( \frac{2}{1.1} \right) + \left( \frac{1}{10} \right) \left( \frac{2}{1.2} \right) + \dots + \left( \frac{1}{10} \right) \left( \frac{2}{2} \right)\).
5. Calculate the sum and round the answer to the nearest hundredth to get the final result.
To approximate the area under the curve y=2/x from 1 to 2 using ten approximating rectangles of equal widths and right endpoints, we can use the Riemann Sum method.
First, we need to calculate the width of each rectangle. Since there are ten rectangles, the interval from 1 to 2 will be divided into ten equal parts.
Width of each rectangle = (2 - 1) / 10 = 1/10
Now, we need to calculate the height of each rectangle. The height will be the value of the function at the right endpoint of each rectangle.
For the first rectangle, the right endpoint is 1 + (1/10) = 1.1. So, the height of the first rectangle is 2/(1.1).
For the second rectangle, the right endpoint is 1.1 + (1/10) = 1.2. So, the height of the second rectangle is 2/(1.2).
And so on, we can calculate the height of each rectangle till the tenth rectangle, where the right endpoint will be 1.9.
Once we have the width and height of each rectangle, we can calculate the area of each rectangle by multiplying its width and height. Finally, we add up the areas of all the rectangles to get the approximate area under the curve.
Area ≈ width * (height 1 + height 2 + ... + height 10)
So, the calculation would look like:
Area ≈ (1/10) * (2/(1.1) + 2/(1.2) + ... + 2/(1.9))
Evaluating this expression will give you the approximate area under the curve.
Do the calculations and round your answer to the nearest hundredth.