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 \(R .\) Repeat Exercise 1\((b)\) for RRAM and MRAM.

Short answer

The region enclosed by the graph of the function and the x-axis has been sketched and partitioned into four subintervals. Rectangles have been drawn to illustrate LRAM, RRAM, and MRAM methods of approximating the area under the curve. LRAM, RRAM and MRAM offer different visual representations of how areas under a curve can be approximated by rectangles. These methods use the left endpoint, right endpoint, and midpoint of the divisions, respectively.

Step-by-step solution

Sketching the region

First of all, it's necessary to identify the region \(R\). To do that, plot the function \(y=2x - x^2\) for the range \(0 \leq x \leq 2\). This will yield a parabolic curve opening downwards. The region \(R\) to be considered will be the one enclosed between the graph of the function and the x-axis.

Partitioning the Interval

Partition the given interval [0,2] into four equal sub-intervals, and denote them as [0,0.5], [0.5,1], [1,1.5] and [1.5,2]. These are the x-values that will be used to draw the rectangles in the following steps.

Constructing the Rectangles for LRAM

Now we apply the Left Rectangular Approximation Method (LRAM). Start by choosing the left endpoint of each sub-interval to draw four rectangles for each of these sub-intervals. Colour in those rectangles to highlight them.

Constructing the Rectangles for RRAM

Next, construct the rectangles using Right Rectangular Approximation Method (RRAM). The process is similar to LRAM, but now choose the right endpoint of each sub-interval. Draw rectangles using these points and then colour them.

Constructing Rectangles for MRAM

Lastly, we demonstrate Midpoint Rectangular Approximation Method (MRAM). For each sub-interval, choose the midpoint. Create and colour rectangles using these points.