Use left and right endpoints and the given number of rectangles to find two approximations of the area of the region between the graph of the function and the x-axis over the given interval.f(x) = 2x + 9, [0, 2], 4 rectangles
Question
Use left and right endpoints and the given number of rectangles to find two approximations of the area of the region between the graph of the function and the x-axis over the given interval.f(x) = 2x + 9, [0, 2], 4 rectangles
Solution
Sure, let's find the area under the curve using both left and right endpoints.
First, we need to find the width of each rectangle. The width (Δx) is (b - a) / n, where a and b are the limits of integration and n is the number of rectangles. Here, a = 0, b = 2, and n = 4. So, Δx = (2 - 0) / 4 = 0.5.
The x-values of the endpoints of the rectangles are then: 0, 0.5, 1, 1.5, and 2.
- Left endpoint approximation:
We use the left endpoints of each subinterval to form the rectangles. The height of each rectangle is the value of the function at the left endpoint. So, the areas of the rectangles are:
Rectangle 1: f(0) * Δx = (20 + 9) * 0.5 = 4.5 Rectangle 2: f(0.5) * Δx = (20.5 + 9) * 0.5 = 5 Rectangle 3: f(1) * Δx = (21 + 9) * 0.5 = 5.5 Rectangle 4: f(1.5) * Δx = (21.5 + 9) * 0.5 = 6
The total area is the sum of these, which is 4.5 + 5 + 5.5 + 6 = 21.
- Right endpoint approximation:
We use the right endpoints of each subinterval to form the rectangles. The height of each rectangle is the value of the function at the right endpoint. So, the areas of the rectangles are:
Rectangle 1: f(0.5) * Δx = (20.5 + 9) * 0.5 = 5 Rectangle 2: f(1) * Δx = (21 + 9) * 0.5 = 5.5 Rectangle 3: f(1.5) * Δx = (21.5 + 9) * 0.5 = 6 Rectangle 4: f(2) * Δx = (22 + 9) * 0.5 = 6.5
The total area is the sum of these, which is 5 + 5.5 + 6 + 6.5 = 23.
So, the left endpoint approximation of the area under the curve is 21, and the right endpoint approximation is 23.
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