Final answer:
To estimate the area under the graph of f(x) = 1/x+4 over the interval [2,4] using five approximating rectangles and right endpoints, we can use the Right Riemann Sum method. By dividing the interval into five equal subintervals and calculating the heights of the rectangles using the right endpoints, we can find the area of each rectangle and sum them up to find the estimated area.
Step-by-step explanation:
To estimate the area under the graph of f(x) = 1/x+4 over the interval [2,4] using five approximating rectangles and right and left endpoints, we can use the Right and Left Riemann Sums.
Right Riemann Sum:
Step 1: Divide the interval [2,4] into five equal subintervals.
Step 2: Calculate the width of each rectangle: Δx = (b - a)/n = (4 - 2)/5 = 0.4
Step 3: Calculate the heights of the rectangles using the right endpoints of each subinterval. The heights are: f(2.4) = 1/2.4+4 = 0.357, f(2.8) = 1/2.8+4 = 0.294, f(3.2) = 1/3.2+4 = 0.25, f(3.6) = 1/3.6+4 = 0.222, f(4) = 1/4+4 = 0.2.
Step 4: Calculate the area of each rectangle: A = base × height=Δx × f(x). The areas are: A1 = 0.4 × 0.357 = 0.1428, A2 = 0.4 × 0.294 = 0.1176, A3 = 0.4 × 0.25 = 0.1, A4 = 0.4 × 0.222 = 0.0888, A5 = 0.4 × 0.2 = 0.08.
Step 5: Sum up the areas of all the rectangles: 0.1428 + 0.1176 + 0.1 + 0.0888 + 0.08 = 0.5292.
Therefore, the estimated area under the graph of f(x) = 1/x+4 over the interval [2,4] using five approximating rectangles and right endpoints is 0.5292.