Problem 47 - Calculus for Biology and Medicine

Question

As definite integrals. Note that (1) \(P=\left[x_{0}, x_{1}, \ldots, x_{n}\right]\) is a partition of the indicated interval, (2) \(c_{k} \in\left[x_{k-1}, x_{k}\right]\), and (3) \(\Delta x_{k}=x_{k}-x_{k-1}\). \(\lim _{\|P\| \rightarrow 0} \sum_{k=1}^{n} e^{c_{k}} \Delta x_{k}\), where \(P\) is a partition of \([-5,2]\)

Step-by-Step Solution

Answer

The limit of the Riemann sum as \(\|P\| \rightarrow 0\) is \(e^2 - e^{-5}\).

Step 1: Understand the Limit Definition of the Integral

The expression given is a Riemann sum, which is used to approximate the definite integral of a function over a particular interval. This sum approaches the definite integral as the partition becomes finer (\(\|P\|\to0\)), where \(\|P\|\) represents the norm or width of the largest subinterval in the partition.

Step 2: Identify the Function and Interval

The function provided in the sum is \(e^{c_{k}}\), and the interval where the integration is performed is from \(-5\) to \(2\). This suggests that the Riemann sum represents the integral of the function \(f(x) = e^x\) over the interval \([-5, 2]\).

Step 3: Set Up the Definite Integral

To find the integral of \(f(x) = e^x\) over the interval \([-5, 2]\), we can write the definite integral as: \[\int_{-5}^{2} e^x \, dx.\]

Step 4: Calculate the Definite Integral

The integral of \(e^x\) is straightforward, as it is \(e^x\) itself. So, we evaluate the integral from \(-5\) to \(2\) as follows:\[\left[ e^x \right]_{-5}^{2} = e^2 - e^{-5}.\]

Step 5: Compute the Final Answer

Substitute the values to find the result of the integral: \[ e^2 - e^{-5} \]. Calculate these values to express a numerical result if necessary.

Key Concepts

Riemann sum

A Riemann sum is a way to approximate the total area under a curve, also known as the definite integral, of a function over a particular interval. It involves dividing the interval into smaller subintervals, represented by a partition. For each subinterval, we select a sample point and evaluate the function at that point. The output is then multiplied by the width of the subinterval.
- The Riemann sum adds up all these little rectangles, whose heights are determined by the function values at the chosen sample points, and whose widths are the corresponding subinterval widths.
- As the partitions are made finer, meaning the width of the largest subinterval goes towards zero, the approximation improves. This is akin to having more, but smaller rectangles, giving a better approximation of the curve's area.
- The exact area under the curve can be obtained in the limit as the partition becomes infinitely fine, turning the approximation into the integral of the function.

limit definition of the integral

The limit definition of the integral is a mathematical representation of how we define the definite integral of a function, using the concept of limits. This definition formalizes the idea of taking the limit of a Riemann sum as the norm of the partition approaches zero.
- In this context, the norm or width \(\|P\|\) of the partition is the width of the largest subinterval in the partition.
- Essentially, this means we are looking at the sum of areas of rectangles whose number goes to infinity and whose width goes to zero. This sum then converges to the precise area under the curve when viewed as an integral.
- For our example with the function \(f(x) = e^x\) over the interval \([-5, 2]\), the limit definition suggests that as \(\|P\|\) approaches zero, the Riemann sum \(\sum_{k=1}^{n} e^{c_{k}} \Delta x_{k}\) becomes the definite integral \(\int_{-5}^{2} e^x \, dx\).

partition of an interval

Partitioning an interval is a foundational concept when dealing with integration and Riemann sums. It involves dividing a larger interval into smaller, non-overlapping pieces, called subintervals.
- For example, partitioning the interval \([-5, 2]\) means creating subintervals like \([x_0, x_1], [x_1, x_2], \ldots, [x_{n-1}, x_n]\). The endpoints \(x_0=-5\) and \(x_n=2\) mark the beginning and end of the overall interval.
- The partition’s quality is often measured by the norm of the partition, which is determined by the width of the largest subinterval, denoted \(\Delta x_k = x_k - x_{k-1}\).
- In a Riemann sum, each subinterval is used to approximate a portion of the area under the curve by evaluating the function at specific points \(c_k\) in each subinterval. As the partition becomes finer (meaning smaller \(\Delta x_k\)), the Riemann sum becomes a better approximation of the integral of the function over the whole interval.