Question

Exercises \(5-8\) refer to the region \(R\) enclosed between the graph of the function \(y=2 x-x^{2}\) and the \(x\) -axis for \(0 \leq x \leq 2\) . (a) Sketch the region \(R\) . (b) Partition \([0,2]\) into 4 subintervals and show the four rectangles that LRAM uses to approximate the area of \(R .\) Compute the LRAM sum without a calculator

Short answer

The rectangles are at x-values of 0, 0.5, 1.0, and 1.5. After calculating the area of each rectangle and summing them up, the LRAM approximation for the area under the curve from x=0 to x=2 is 1.375.

Step-by-step solution

- Sketch the region

First, based on the function \(f(x) = 2x - x ^2\), sketch the parabola over the range from 0 to 2. This should include the x-axis, acting as the lower boundary for every rectangle in LRAM. Observe that, the graph is a downward-opening parabola that intersects the x-axis at x=0 and x=2, resulting in a shape like a hill.

- Partition into subintervals and identify rectangles

As per the second part of the question, we partition the range [0, 2] into 4 equal subintervals. Each subinterval forms the base of a rectangle. The left end point of each subinterval, when substituted in the function gives the height of the corresponding rectangle. Hence, the rectangles are at x-values of 0, 0.5, 1.0, and 1.5.

- Compute LRAM sum

For each of the rectangles, calculate the area (width*height). Sum these areas to approximate the area under the curve i.e. to compute LRAM. Substitute the x-values of each rectangle into the function to calculate the height of each rectangle. Simplify the sum to get the final approximate area.