Question

Estimate the area under the graph of f(x) = 5 cos(x) from x = 0 to x = π/2 using four approximating rectangles and right endpoints. (Round your answers to four decimal places.)

R4 =

Answer 1

To estimate the area under the graph of f(x) = 5 cos(x) from x = 0 to x = π/2 using four approximating rectangles and right endpoints, we can use the right endpoint rule. The estimate is 0.7408.

Step-by-step explanation:

To estimate the area under the graph of f(x) = 5 cos(x) from x = 0 to x = π/2 using four approximating rectangles and right endpoints, we can use the right endpoint rule. The right endpoint rule states that the height of each rectangle is the value of the function at the right endpoint of the interval, and the width of each rectangle is the width of the interval. In this case, the width of each rectangle is π/8 since there are four rectangles dividing the interval [0, π/2]. So, the area of each rectangle is 5cos(π/8) * (π/8) = 0.1852. Summing up the areas of all four rectangles gives the estimate of the area under the graph:

R4 = 0.1852 + 0.1852 + 0.1852 + 0.1852 = 0.7408 (rounded to four decimal places).

Answer 2

The estimated area under the graph using 4 right rectangles is approximately 4.4058 square units.

Here's how to estimate the area using right Riemann sums with 4 rectangles:

1. Determine the width of each rectangle:

The interval is [0, π/2], so its width is (π/2 - 0) = π/2.

With 4 rectangles, the width of each rectangle is (π/2)/4 = π/8.

2. Identify the right endpoints:

The right endpoints of the rectangles will be at x = π/8, 3π/8, 5π/8, and 7π/8.

3. Evaluate the function at these endpoints:

f(π/8) ≈ 4.7553

f(3π/8) ≈ 4.1213

f(5π/8) ≈ 2.9389

f(7π/8) ≈ 1.3147

4. Calculate the area of each rectangle:

Rectangle 1: width * height = (π/8) * 4.7553 ≈ 1.5460

Rectangle 2: width * height = (π/8) * 4.1213 ≈ 1.3405

Rectangle 3: width * height = (π/8) * 2.9389 ≈ 1.0546

Rectangle 4: width * height = (π/8) * 1.3147 ≈ 0.4547

5. Sum the areas of the rectangles:

R4 = 1.5460 + 1.3405 + 1.0546 + 0.4547 ≈ 4.4058