Problem 38 - Calculus for Biology and Medicine

Question

(a) Assume that \(a>0\). Evaluate \(\int_{0}^{a} x d x\), using the fact that the region bounded by \(y=x\) and the \(x\) -axis between 0 to \(a\) is a triangle. (b) Assume that \(a>0\). Evaluate \(\int_{0}^{a} x d x\) by approximating the region bounded by \(y=x\) and the \(x\) -axis from 0 to \(a\) with rectangles. Use equal subintervals and take right endpoints.

Step-by-Step Solution

Answer

The integral \(\int_{0}^{a} x \, dx\) evaluates to \(\frac{1}{2}a^2\) by both geometry and approximation methods.

Step 1: Analyzing the Geometric Interpretation

The region bounded by \(y = x\), the x-axis, and the lines \(x = 0\) and \(x = a\) forms a right triangle. In a triangle, the area can be calculated using the formula \(\frac{1}{2}\times \text{base}\times \text{height}\). Here, the base is \(a\) (along the x-axis) and the height is also \(a\) (along the y-axis, as \(y = x\) at \(x = a\)). Thus, the area is \(\frac{1}{2}a^2\).

Step 2: Integrating Using Geometry

Since the integral \(\int_{0}^{a} x \, dx\) represents the area of the triangle described, we use the geometric interpretation to find: \(\int_{0}^{a} x \, dx = \frac{1}{2}a^2\).

Step 3: Approximating with Rectangles

To approximate the integral \(\int_{0}^{a} x \, dx\) using rectangles, divide the interval \([0, a]\) into \(n\) subintervals of equal width \(\Delta x = \frac{a}{n}\). Each subinterval can be noted as \([x_{i-1}, x_i]\) with \(x_i = \frac{i \cdot a}{n}\).

Step 4: Function Value at Right Endpoint

For the right-endpoint approximation, evaluate the function at the right endpoint of each subinterval: \(f(x_i) = x_i\). This gives the height of each rectangle as \(\frac{i\cdot a}{n}\).

Step 5: Calculating the Sum of Rectangle Areas

The area of one rectangle using the right endpoint is \(f(x_i)\cdot \Delta x = \frac{i \cdot a}{n} \cdot \frac{a}{n} = \frac{i \cdot a^2}{n^2}\). Summing the areas of all \(n\) rectangles gives: \(\sum_{i=1}^{n} \frac{i \cdot a^2}{n^2}\).

Step 6: Solving the Series

The series becomes: \(\frac{a^2}{n^2} \sum_{i=1}^{n} i = \frac{a^2}{n^2} \cdot \frac{n(n+1)}{2}\) based on the formula for the sum of the first \(n\) integers. Simplifying gives: \(\frac{a^2(n+1)}{2n}\).

Step 7: Taking the Limit as n Approaches Infinity

To find the exact area, take the limit of this approximation as \(n \to \infty\): \(\lim_{n \to \infty} \frac{a^2(n+1)}{2n} = \frac{a^2}{2}\).

Step 8: Conclusion

Both methods—the geometric interpretation and approximation with rectangles—confirm the value of the integral: \(\int_{0}^{a} x \, dx = \frac{1}{2}a^2\).

Key Concepts

Geometric Interpretation in Calculus

The geometric interpretation of integrals is a powerful concept in calculus. It connects algebraic expressions with spatial understanding. In calculus, integrals often represent the area under a curve. For the function \(y = x\) from \(x=0\) to \(x=a\), this area forms a right triangle. Here, the base and height of the triangle are both \(a\). Using the formula for the area of a triangle, \(\frac{1}{2} \times \text{base} \times \text{height}\), the area can be calculated as \(\frac{1}{2}a^2\).
This geometric representation not only makes the concept of the definite integral tangible but also provides a simple visual method to evaluate it. The triangle gives us an intuitive shortcut to determine the value without performing actual integration, demonstrating the deep connection between geometry and calculus.

Right Endpoint Approximation

Right endpoint approximation is a numerical method for estimating the value of a definite integral. This technique involves dividing the area under the curve into rectangles and using the value of the function at the right endpoint of each subinterval to set the height of the rectangle.

Suppose we split the interval \([0, a]\) into \(n\) equal parts. Each part, or subinterval, has a width of \(\Delta x = \frac{a}{n}\). At each right endpoint \(x_i = \frac{i\cdot a}{n}\), the function value \(f(x_i)\) gives the rectangle's height. Hence, each rectangle's area becomes \(f(x_i) \Delta x = \frac{i \cdot a^2}{n^2}\).
This approximation yields a sum of all rectangular areas, \(\sum_{i=1}^{n} \frac{i \cdot a^2}{n^2}\), which approximates the integral. By increasing \(n\), the rectangles better approximate the true area under the curve.

Sum of Series in Calculus

The final step in using right endpoint approximation involves calculating the sum of the series. In calculus, a commonly used result is the sum of the first \(n\) natural numbers: \(1 + 2 + 3 + \, \ldots \, + n = \frac{n(n+1)}{2}\).

Applying this to our right endpoint approximation, we get:

• The sum of individual rectangle areas is \(\sum_{i=1}^{n} \frac{i \cdot a^2}{n^2}\).
• This becomes \(\frac{a^2}{n^2} \sum_{i=1}^{n} i = \frac{a^2}{n^2} \cdot \frac{n(n+1)}{2}\).
• Upon simplification, this results in \(\frac{a^2(n+1)}{2n}\).

Finally, to determine the precise integral value, take the limit as \(n\) approaches infinity. The series simplifies to \(\frac{a^2}{2}\), matching the area from geometric interpretation.