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 \(R .\) Make a conjecture about the area of the region \(R .\)

Short answer

The region R is the area enclosed by the curve \(y=2x-x^2\) and the x-axis between x=0 and x=2. It's partitioned into four subintervals for the Left Riemann Sum approximation. The area is more than the sum of areas of the rectangles obtained.

Step-by-step solution

Sketch the region R

First, plot the equation \(y=2x-x^2\) on a graph. This is a downward-opening parabola with roots at x=0 and x=2 and a peak at \(x=1\). The area of region R would be the one under the curve between x=0 and x=2.

Partition the interval

Now, divide the x-axis interval [0,2] into four equal parts, namely [0,0.5], [0.5,1], [1,1.5] and [1.5,2]. These subintervals will be used for the LRAM approximation.

Left Riemann Sum Method (LRAM)

For each subinterval, draw a rectangle that touches the curve at the left endpoint. In this case, for the interval [0,0.5], the height of the rectangle would be the function's value at \(x=0\) (which is 0); for [0.5,1], the height would be the value at \(x=0.5\), and so on. Sum up the areas of these rectangles to approximate the area under the curve.

Make a conjecture

Add up the areas of all the rectangles to get an approximate of the area for region R. This sum is much likely to be lesser than the real area since all rectangles were under the curve. Thus, the real area of region R may be somewhat greater than the approximate area calculated using LRAM.

Key concepts

Definite Integrals

A definite integral is a crucial concept in calculus that represents the accumulation of quantities, such as areas. It is the limit of a sum of areas of rectangles, known as a Riemann sum, as the width of those rectangles approaches zero. When we talk about the area under a curve between two points on the x-axis, we often refer to this area as a definite integral. Mathematically, the definite integral of a function from a to b is denoted as \( \[ \int_a^b f(x) \, dx \] \), where \( f(x) \) is the function being integrated and \( a \) and \( b \) are the limits of integration. In the context of our original exercise, the definite integral would give us the exact area of region R, the area under the parabola \( y = 2x - x^2 \) from \( x = 0 \) to \( x = 2 \).

Area Under a Curve

Understanding the concept of the area under a curve is pivotal in calculus. It allows us to quantify the space enclosed between the graph of a function and the x-axis over a particular interval. This idea is not just limited to simple geometric shapes but extends to any function graphed on a coordinate plane. The greater the function's values over an interval, the larger the area under the curve. In practical terms, this area can signify anything from physical distances to probabilities in statistics. In the case of our parabolic function \( y = 2x - x^2 \), the region below the curve and above the x-axis from \( x = 0 \) to \( x = 2 \) — this is the area we aim to determine using LRAM or by calculating the definite integral.

Riemann Sums

Riemann sums are used to approximate the area under a curve—basically, the groundwork behind integration. There are different types of Riemann sums depending on where you choose the height of the rectangles: left, right, midpoints, or even a random point within each interval. These sums are calculated by adding up the areas of the chosen rectangles. Specifically, a Left Riemann Sum (LRAM) utilizes the leftmost point of each subinterval to determine the height of the rectangle. Returning to our exercise, we applied LRAM to estimate the area of region R, seeing that rectangles are formed at the left endpoints of subintervals within the given range. While LRAM is a valuable tool for approximation, it's not as accurate as the exact calculation performed through the definite integral.

Parabolic Functions

Parabolic functions are quadratic functions of the form \( f(x) = ax^2 + bx + c \), whose graphs are called parabolas. Depending on the coefficient \( a \), the parabola opens upwards (for \( a > 0 \)) or downwards (for \( a < 0 \)). Characteristics of parabolas, such as the vertex, axis of symmetry, and roots (zeros), are vital in understanding their behaviors. The function given in the original problem, \( y = 2x - x^2 \), is a downward-opening parabola with the vertex, which happens to be the maximum point on the graph, at \( x = 1 \). When graphed, this parabola intersects the x-axis at its roots at \( x = 0 \) and \( x = 2 \), forming the base for the region R that we are interested in. Interestingly, the properties of parabolas allow for various applications in physics, engineering, and economics, illustrating the real-world relevance of understanding these functions.