Approximate the area under the curve and above the x-axis using n rectangles. Let the height of each rectangle be given by the value of the function at the right side of the rectangle.
f(x)=3/x from x=2 to x=6; n=4
for x = 3,4,5,6
f(x) = 1,3/4,3/5,1/2
Now just add up all the rectangles of width 1.
I'm lost here. I think i'm adding wrong numbers or something else here.
I guess you must be. Draw a diagram, and draw 4 rectangles of width 1, from
x=2-3, 3-4, 4-5, 5-6 high enough so that the top right corner of each rectangle intersects the graph of y=3/x.
rectangle are thus
1 x 1
1 x 3/4
1 x 3/5
1 x 3/6
Approximate the area under the curve by adding up the areas of all the rectangles. That is,
1 + 3/4 + 3/5 + 1/2 = 57/20 = 2.85
Well, let's examine the problem carefully. We need to approximate the area under the curve of f(x) = 3/x, above the x-axis, using 4 rectangles.
To approximate the area, we'll divide the interval [2,6] into 4 equal subintervals. Each subinterval will have a width of (6-2)/4 = 1.
Now, let's find the right side of each rectangle. The right side of the first rectangle will be x = 2 + 1 = 3. The right side of the second rectangle will be x = 3 + 1 = 4. The right side of the third rectangle will be x = 4 + 1 = 5. And finally, the right side of the fourth rectangle will be x = 5 + 1 = 6.
Next, we'll evaluate the function f(x) = 3/x at each right side value to find the height of each rectangle.
For the first rectangle: Height = f(3) = 3/3 = 1.
For the second rectangle: Height = f(4) = 3/4 = 0.75.
For the third rectangle: Height = f(5) = 3/5 = 0.6.
For the fourth rectangle: Height = f(6) = 3/6 = 0.5.
Now, we can calculate the area of each rectangle by multiplying its width by its height.
Area of the first rectangle = 1 * 1 = 1.
Area of the second rectangle = 1 * 0.75 = 0.75.
Area of the third rectangle = 1 * 0.6 = 0.6.
Area of the fourth rectangle = 1 * 0.5 = 0.5.
Finally, we sum up the areas of all 4 rectangles to get an approximate total area under the curve and above the x-axis: 1 + 0.75 + 0.6 + 0.5 = 2.85 square units.
So, the approximate area under the curve f(x) = 3/x and above the x-axis using 4 rectangles is approximately 2.85 square units. Now that's geometry with a rectangular twist!
To approximate the area under the curve using n rectangles, we can use the method of Riemann sums. In this case, we will use the right Riemann sum, where the height of each rectangle is given by the value of the function at the right side of the rectangle.
First, let's determine the width of each rectangle. Since we are dividing the interval from x=2 to x=6 into 4 rectangles, the width of each rectangle will be:
width = (end point - start point) / n
= (6 - 2) / 4
= 4 / 4
= 1
Next, we will calculate the height of each rectangle using the right side of each rectangle. For each rectangle, we will evaluate the function at the right side of the rectangle and multiply it with the width of the rectangle.
The coordinates of the right side of each rectangle will be:
x1 = 2 + 1 = 3
x2 = 3 + 1 = 4
x3 = 4 + 1 = 5
x4 = 5 + 1 = 6
Now, let's evaluate the function at each of these x values and calculate the height of each rectangle:
h1 = f(3) = 3/3 = 1
h2 = f(4) = 3/4 = 0.75
h3 = f(5) = 3/5 = 0.6
h4 = f(6) = 3/6 = 0.5
Finally, we calculate the area of each rectangle by multiplying the height with the width, and then sum up the areas of all the rectangles to get the approximate area under the curve:
Area = h1 * width + h2 * width + h3 * width + h4 * width
= 1 * 1 + 0.75 * 1 + 0.6 * 1 + 0.5 * 1
= 1 + 0.75 + 0.6 + 0.5
= 2.85
Therefore, using 4 rectangles, the approximate area under the curve and above the x-axis is 2.85.