Question

In Exercises \(9-12,\) use RAM to estimate the area of the region enclosed between the graph of \(f\) and the \(x\) -axis for \(a \leq x \leq b\) . $$f(x)=x^{2}-x+3, \quad a=0, \quad b=3$$

Short answer

The estimated area under the curve \(f(x) = x^{2} - x + 3\) from \(x = 0\) to \(x = 3\) using the rectangular approximation method is 11 square units.

Step-by-step solution

Function Evaluation

Evaluate the function \( f(x) = x^{2} - x + 3 \) at \( x = a \) and \( x = b \), which respectively results in \( f(0) = 3 \) and \( f(3) = 9 \).

Partition the Interval

Given that \( a = 0 \) and \( b = 3 \), it's necessary to partition this interval into subintervals. In this case, choose 3 subintervals for simplicity, so each interval will have a length of \( (3-0)/3 = 1 \). The end-points for these intervals are \( x = 0, 1, 2, 3 \).

Compute Area of Each Rectangle

For each subinterval, compute the area of the rectangle under the curve, which can be estimated using the height times the width of the rectangle. The height is the value of the function at that interval, and the width is the length of the interval which is 1. Therefore, the areas for these three subintervals are \( f(0) * 1 = 3, \: f(1) * 1 = 3, \: f(2) * 1 = 5 \).

Sum the Areas

Lastly, to estimate the total area under the curve on the given interval, sum up the areas of each individual rectangle obtained in Step 3. So, the total area is \(3 + 3 + 5 = 11\) square units.

Key concepts

Definite Integrals

Definite integrals are foundational to calculus and represent the accumulation of quantities, such as area, over an interval. Formally, the definite integral of a function from a to b is the limit of the sum of areas of rectangles (Riemann sums) as the number of rectangles approaches infinity. In simpler terms, if you want to know the total area under a curve between two points on the x-axis, you calculate the definite integral of the function between those points. This process turns the 'slicing' of areas into a precise value that represents the exact accumulation of space between the curve and the x-axis.

Area Under Curve

The concept of finding the area under a curve is at the heart of integration. Integrals allow us to calculate the space enclosed between a curve represented by a function, the x-axis, and the vertical lines defined by the interval's endpoints. This can be visualized as the total area contained within the graphed function over a specific range. For example, the function \( f(x) = x^2 - x + 3 \) evaluated between the points \( a = 0 \) and \( b = 3 \) encapsulates a specific shape on the graph, where the integral quantifies the space or 'the total area' of that shape.

Function Evaluation

Evaluating a function is a fundamental process in mathematics, which involves finding the output of a function for a particular input. When we evaluate the function \( f(x) \) at various points within a specific interval, it's like taking snapshots of the function's height at different stages along the x-axis. The function \( f(x) = x^2 - x + 3 \) evaluated at the endpoints of the interval \( [0,3] \) gives us important values needed to approximate the area under the curve using Riemann sums. These values become the heights of rectangles that we use to build a visual approximation of the area.

Interval Partitioning

Interval partitioning describes a method of dividing the domain of a function into smaller subintervals, where each subinterval can be used to create a rectangle underneath a curve. This technique is used in the approximation methods for integrals, such as Riemann sums. The finer the partition (that is, the more subintervals we create), the closer our approximation will be to the true area under the curve. For instance, partitioning the interval from \( a = 0 \) to \( b = 3 \) into three equal parts provides a groundwork for us to construct rectangles that touch the function at specific points, enabling us to estimate the area under \( f(x) \) over that interval.