Question

approximate the area under the function f(x)=4/x on the interval [1,5] using 8 right-sided rectangles. give your answer as a fraction

Answer 1

The approximate area under the function f(x) = 4/x from x=1 to x=5 using 8 right-sided rectangles is found by calculating the sum of the areas of these rectangles. Each rectangle has a width of 0.5 and the height is determined by the function value at the right endpoints of the subintervals. The total area is the sum of all these individual rectangle areas.

Step-by-step explanation:

To approximate the area under the function f(x) = 4/x on the interval [1,5] using 8 right-sided rectangles, we will use the method of right Riemann sums. The width of each rectangle will be the length of the interval divided by the number of rectangles, which is ∆x = (5-1)/8 = 0.5. We then evaluate the function at the right endpoints of these intervals for the heights of the rectangles:

• Rectangle 1: x = 1.5, height = f(1.5) = 4/1.5
• Rectangle 2: x = 2, height = f(2) = 4/2
• Rectangle 3: x = 2.5, height = f(2.5) = 4/2.5
• Rectangle 4: x = 3, height = f(3) = 4/3
• Rectangle 5: x = 3.5, height = f(3.5) = 4/3.5
• Rectangle 6: x = 4, height = f(4) = 4/4
• Rectangle 7: x = 4.5, height = f(4.5) = 4/4.5
• Rectangle 8: x = 5, height = f(5) = 4/5

The total approximate area is calculated by summing up the area of each rectangle:

Total Area = ∆x(f(1.5) + f(2) + f(2.5) + f(3) + f(3.5) + f(4) + f(4.5) + f(5))

Plugging in the values, we get:

Total Area = 0.5(4/1.5 + 4/2 + 4/2.5 + 4/3 + 4/3.5 + 4/4 + 4/4.5 + 4/5)

This simplifies to:

Total Area = 2(8/3 + 2 + 8/5 + 4/3 + 8/7 + 1 + 8/9 + 4/5)

Now add all the fractions to get the final area as a fraction.

Answer 2

The area under the function f(x)=4/x on the interval [1,5] using 8 right-sided rectangles can be approximated by dividing the interval into parts, calculating the height at the right side of each rectangle, and then summing up all the individual areas.

Step-by-step explanation:

To approximate the area under the function f(x)=4/x on the interval [1,5] using 8 right-sided rectangles, we first need to split the interval into 8 equal parts. The width of each rectangle (Δx) will be (5-1)/8 = 0.5. The height of the rectangles will be determined by the function value at the right end of each subinterval. We will use these heights and the uniform width to calculate the approximate area.

1. For the first rectangle, from x = 1 to x = 1.5, the height is f(1.5) = 4/1.5.
2. For the second rectangle, from x = 1.5 to x = 2, the height is f(2) = 4/2.
3. Continue this process until the eighth rectangle, from x = 4.5 to x = 5, with height f(5) = 4/5.

The approximate total area (A) can be calculated by summing the areas of all the rectangles: A = Δx(f(1.5) + f(2) + f(2.5) + ... + f(5)). Substituting the values and performing the arithmetic gives:

A = 0.5*(4/1.5 + 4/2 + 4/2.5 + 4/3 + 4/3.5 + 4/4 + 4/4.5 + 4/5).

After calculating, we can express the approximate area as a fraction.