Let R be the plane region bounded by x=1, x=5, the x-axis and y=1/x. Approximate the are of R using 8 right rectangles.
To approximate the area of the region R using right rectangles, we can use the right rectangle method or the Riemann sum.
Here's how you can use the right rectangle method with 8 rectangles to approximate the area of R:
1. First, divide the interval [1, 5] into 8 subintervals of equal width. To do this, calculate the width of each subinterval by dividing the total width of the interval by the number of rectangles:
width = (5 - 1) / 8 = 0.5
2. Next, choose a height for each rectangle. The height of each rectangle is determined by the function y = 1/x. To approximate the height, choose a representative x-value within each subinterval and evaluate the function at that x-value. For the right rectangle method, we choose the right endpoint of each subinterval as the x-value.
3. Calculate the area of each rectangle by multiplying the width of the rectangle by its height.
4. Add up the areas of all the rectangles to get the total approximate area of R.
Let's go through the calculations:
Subinterval 1: x = 1, height = 1/1 = 1
Area of rectangle 1: (0.5) * (1) = 0.5
Subinterval 2: x = 1.5, height = 1/1.5 ≈ 0.6667
Area of rectangle 2: (0.5) * (0.6667) = 0.3334
Subinterval 3: x = 2, height = 1/2 = 0.5
Area of rectangle 3: (0.5) * (0.5) = 0.25
... Repeat this process for subintervals 4-8 ...
Subinterval 4: x = 2.5, height = 1/2.5 ≈ 0.4
Area of rectangle 4: (0.5) * (0.4) = 0.2
Subinterval 5: x = 3, height = 1/3 ≈ 0.3333
Area of rectangle 5: (0.5) * (0.3333) = 0.16665
Subinterval 6: x = 3.5, height = 1/3.5 ≈ 0.2857
Area of rectangle 6: (0.5) * (0.2857) ≈ 0.14285
Subinterval 7: x = 4, height = 1/4 = 0.25
Area of rectangle 7: (0.5) * (0.25) = 0.125
Subinterval 8: x = 4.5, height = 1/4.5 ≈ 0.2222
Area of rectangle 8: (0.5) * (0.2222) ≈ 0.1111
Now, add up the areas of all the rectangles:
Total approximate area of R ≈ 0.5 + 0.3334 + 0.25 + 0.2 + 0.16665 + 0.14285 + 0.125 + 0.1111 = 1.729
Therefore, the approximate area of R using 8 right rectangles is approximately 1.729 square units.