Problem 16 - Biocalculus Calculus for the Life Sciences

Question

\(15-17\) Use Definition 2 to find an expression for the area under the graph of \(f\) as a limit. Do not evaluate the limit. $$f(x)=x^{2}+\sqrt{1+2 x}, 4 \leqq x \leq 7$$

Step-by-Step Solution

Answer

Area is expressed as a limit: \( A = \lim_{n \to \infty} \sum_{i=1}^{n} \left((4 + i \frac{3}{n})^2 + \sqrt{1 + 2(4 + i \frac{3}{n})}\right) \frac{3}{n} \).

Step 1: Understand the Problem

We need to express the area under the curve of the given function, which is between the limits 4 and 7, as a limit. We will use the idea of a Riemann sum to do this.

Step 2: Partition the Interval

Consider partitioning the interval [4, 7] into 'n' equal subintervals. The width of each subinterval, \( \Delta x \), is \( \frac{7 - 4}{n} = \frac{3}{n} \).

Step 3: Determine Sample Points

Select sample points \( x_i^* \) within each subinterval \([x_{i-1}, x_i]\). We typically use the right endpoint of each subinterval, thus \( x_i^* = 4 + i\Delta x = 4 + i \frac{3}{n} \).

Step 4: Set Up the Riemann Sum

The Riemann sum approximating the area under the curve is given by: \[ S_n = \sum_{i=1}^{n} f(x_i^*) \Delta x \]where \( f(x) = x^2 + \sqrt{1+2x} \) and \( \Delta x = \frac{3}{n} \). Substituting the expressions for \( f(x) \) and \( x_i^* \), we have:\[ S_n = \sum_{i=1}^{n} \left(\left(4 + i \frac{3}{n}\right)^2 + \sqrt{1 + 2\left(4 + i \frac{3}{n}\right)}\right) \frac{3}{n} \]

Step 5: Express the Area as a Limit

The exact area under the curve is obtained in the limit as the number of subintervals \( n \) approaches infinity. Therefore, the area \( A \) is given by:\[ A = \lim_{n \to \infty} \sum_{i=1}^{n} \left(\left(4 + i \frac{3}{n}\right)^2 + \sqrt{1 + 2\left(4 + i \frac{3}{n}\right)}\right) \frac{3}{n} \]

Key Concepts

Area Under a Curve

The concept of finding the area under a curve is fundamental in calculus and helps us understand accumulated values like distance, time, and more. To determine the area under the curve of a function between two points, we use integrals. In this exercise, we look at the function \( f(x) = x^2 + \sqrt{1 + 2x} \) between \( x = 4 \) and \( x = 7 \). The goal is to express the total area beneath this curve as the limit of a sum of small areas. These small areas are vertical slices under the curve, each approximating the area contained between four and seven. This process sums up the height of the curve at different points, multiplied by the width of each slice, and goes to the limit as these slices get thinner to provide an accurate area.

Limit of a Function

The limit of a function describes the value that a function approaches as the input approaches a certain point. In the context of Riemann sums, limits are essential as they help transition from an approximation to an exact value. When dealing with areas under curves, we approximate the area by dividing it into multiple subareas and adding them up. The precise area is found by taking the limit of these sums as the number of subintervals, \( n \), becomes infinitely large. This means the width of each subinterval \( \Delta x \) becomes infinitesimally small, allowing the Riemann sum to converge to the true value of the integral. This limit process provides us with an exact area under the curve, capturing all the intricacies of the curve's shape.

Subinterval Partitioning

Subinterval partitioning involves dividing the interval over which we are finding the area into smaller, manageable intervals or sections. This technique simplifies the process of calculating areas under curves. For the given function, the interval \([4, 7]\) was divided into \( n \) equal parts, each with a width of \( \Delta x = \frac{3}{n} \). Once partitioned, each subinterval’s contribution to the total area is represented through a sample point, often chosen as the right endpoint \( x_i^* = 4 + i \frac{3}{n} \).

• The *starting point* is 4.
• The *ending point* is 7.
• Each partition contributes a small rectangular area to the total sum.

By choosing better subinterval partitions and taking the limit as \( n \to \infty \), the approximation becomes exact, and the sum of all small contributions equals the desired area under the curve.