Problem 21 - Biocalculus Calculus for the Life Sciences

Question

21\. (a) Express the area under the curve \(y=x^{3}\) from 0 to 2 as a limit. (b) Use a computer algebra system to find the sum in your expression from part (a). (c) Evaluate the limit in part (a).

Step-by-Step Solution

Answer

The area is 4.

Step 1: Understand the problem

We need to express the area under the curve \(y=x^3\) from 0 to 2 as a limit. This involves setting up an integral that represents the area and then expressing it as a limit of a sum.

Step 2: Set up the Riemann sum

To express the integral as a limit, divide the interval [0, 2] into \(n\) equal subintervals, each of width \(\Delta x = \frac{b-a}{n} = \frac{2-0}{n} = \frac{2}{n}\). The endpoints of subintervals can be expressed as \(x_i = 0 + i\Delta x = \frac{2i}{n}\), for \(i = 0, 1, 2, \, ..., n\). The height of each rectangle is \(f(x_i^*)\), where \(x_i^*\) is any point in the subinterval, usually taken as the right end \(x_i\). Therefore, the Riemann sum is \(\sum_{i=1}^{n} f\left(\frac{2i}{n}\right)\cdot \frac{2}{n}\). Here, \(f(x) = x^3\), the sum becomes \(\sum_{i=1}^{n} \left(\frac{2i}{n}\right)^3 \cdot \frac{2}{n}\).

Step 3: Simplify the expression

The Riemann sum can be rewritten as \(\frac{2}{n} \sum_{i=1}^{n} \left(\frac{2i}{n}\right)^3 = \frac{2}{n} \cdot \frac{8}{n^3} \sum_{i=1}^{n} i^3 = \frac{16}{n^4} \sum_{i=1}^{n} i^3\).

Step 4: Express as a limit

The area under the curve from 0 to 2 can be expressed as a limit: \(\lim_{n \to \infty} \frac{16}{n^4} \sum_{i=1}^{n} i^3\). To evaluate this limit, we use the formula for the sum of cubes: \(\sum_{i=1}^{n} i^3 = \left(\frac{n(n+1)}{2}\right)^2\). Substitute this formula into the limit: \(\lim_{n \to \infty} \frac{16}{n^4} \cdot \left(\frac{n(n+1)}{2}\right)^2\).

Step 5: Simplify the limit expression

The expression becomes \(\lim_{n \to \infty} \frac{16}{n^4} \cdot \frac{n^2(n+1)^2}{4}\). This further simplifies to \(\lim_{n \to \infty} \frac{16n^2(n^2 + 2n + 1)}{4n^4}= 4 \cdot \lim_{n \to \infty} \frac{n^4 + 2n^3 + n^2}{n^4}\).

Step 6: Evaluate the limit

Split the limit into separate terms: \(4 \cdot (1 + \frac{2}{n} + \frac{1}{n^2})\). Taking the limit as \(n \to \infty\), terms containing \(\frac{2}{n}\) and \(\frac{1}{n^2}\) approach zero. Thus, the limit evaluates to \(4\).

Step 7: Verify using a computer algebra system

To confirm our calculations, use a computer algebra system like Wolfram Alpha or MATLAB. Input the Riemann sum expression: \(\sum_{i=1}^{n} \left(\frac{2i}{n}\right)^3 \cdot \frac{2}{n}\) and evaluate it directly for large values of \(n\). The result should also approach \(4\).

Key Concepts

Riemann Sum

The Riemann Sum is a foundational concept in Integral Calculus that helps approximate the area under a curve. It does this by summing up the areas of a series of rectangles under the curve. The more rectangles we use, the closer we get to the actual area.
For the function given, \(y = x^3\), the process starts by dividing the interval [0, 2] into