Question

Multiple Choice If an MRAM sum with four rectangles of equal width is used to approximate the area enclosed between the \(x\) -axis and the graph of \(y=4 x-x^{2},\) the approximation is (A) 10 (B) 10.5 (C) \(10 . \overline{6}\) (D) 10.75 (E) 11

Short answer

None of the answer choices provided match the solution. There might be a mistake in the question or the provided options.

Step-by-step solution

Identify the interval

The problem doesn't specifically mention the interval. However, we know that the graph of the polynomial \(y = 4x -x^{2}\) starts from x=0 and ends at x=4, because that's where the equation reaches its maximum value. So, the interval is [0, 4]. The width of each rectangle (delta x) would be (4 - 0) / 4 = 1.

Determine the heights

Midpoint Riemann Approximation Method (MRAM) uses the value of the function at the midpoint of each interval as the height for the rectangles. So we plug the midpoints of each interval into the function to get the heights. The midpoints of our intervals are 0.5, 1.5, 2.5 and 3.5. Plugging these into the equation we get the heights as follows: for x = 0.5, height, \(h_{1} = 4(0.5) - (0.5)^{2} = 1.75\); For x = 1.5, height, \(h_{2} = 4(1.5) - (1.5)^2 = 4.25\); For x = 2.5, height, \(h_{3} = 4(2.5) - (2.5)^2 = 4.75\); For x = 3.5, height, \(h_{4} = 4(3.5) - (3.5)^2 = 4.25\)

Calculate the area

The area under the curve is found by sum of the areas of the individual rectangles. So, the total area = width * sum of the heights = 1 * (1.75 + 4.25 + 4.75 + 4.25) = 15.