Problem 36 - Calculus

Question

Many computer algebra systems permit the evaluation of Riemann sums for left end point, right end point, or midpoint evaluations of the function. Using such a system evaluate the 10-subinterval Riemann sums using left end point, right end point, and midpoint evaluations. $$ \int_{0}^{1} \tan x d x $$

Step-by-Step Solution

Answer

Left endpoint: ~0.1618, Right endpoint: ~0.2449, Midpoint: ~0.2023.

Step 1: Divide the interval into subintervals

Since we want to use 10 subintervals, divide the interval \([0, 1]\) into 10 equal parts. Each subinterval will have a width of \(\Delta x = \frac{1-0}{10} = 0.1\).

Step 2: Left Endpoint Evaluation

For the left endpoint evaluation of the Riemann sum, we use the left end of each subinterval. The left endpoints for these subintervals are \(x_0 = 0, x_1 = 0.1, x_2 = 0.2, \ldots, x_9 = 0.9\). Calculate the sum: \[\sum_{i=0}^{9} \tan(x_i) \cdot \Delta x = \sum_{i=0}^{9} \tan(i \cdot 0.1) \cdot 0.1.\]

Step 3: Right Endpoint Evaluation

For the right endpoint evaluation, we use the right end of each subinterval. The right endpoints are \(x_1 = 0.1, x_2 = 0.2, \ldots, x_{10} = 1.0\). Calculate the sum: \[\sum_{i=1}^{10} \tan(x_i) \cdot \Delta x = \sum_{i=1}^{10} \tan(i \cdot 0.1) \cdot 0.1.\]

Step 4: Midpoint Evaluation

For midpoint evaluation, we use the midpoint of each subinterval. The midpoints are \(x_{0.5} = 0.05, x_{1.5} = 0.15, \ldots, x_{9.5} = 0.95\). Calculate the sum: \[\sum_{i=0}^{9} \tan(x_{i+0.5}) \cdot \Delta x = \sum_{i=0}^{9} \tan((i + 0.5) \cdot 0.1) \cdot 0.1.\]

Step 5: Evaluate Riemann Sums Using a Calculator

Evaluate each sum from steps 2, 3, and 4 using a computer algebra system or a calculator that can compute trigonometric functions. Calculate: - Left endpoint: \(\sum_{i=0}^{9} \tan(i \cdot 0.1) \cdot 0.1\)- Right endpoint: \(\sum_{i=1}^{10} \tan(i \cdot 0.1) \cdot 0.1\)- Midpoint: \(\sum_{i=0}^{9} \tan((i + 0.5) \cdot 0.1) \cdot 0.1\)

Key Concepts

Integration

Integration is a powerful concept in calculus that helps us find the total accumulation of quantities. It can be understood as the opposite process of differentiation. When you integrate a function over a certain interval, you're essentially adding up infinite tiny pieces under the curve of the function. This is extremely useful in physics, engineering, and economics for finding quantities like area, volume, and total change.
For instance, when integrating the function \( \tan(x) \) from \(0\) to \(1\), you compute the area under its curve within this interval. Although finding the exact integral of \( \tan(x) \) analytically might be challenging, especially over finite intervals, numerical methods can approximate these integrals efficiently.

Numerical Methods

Numerical methods come into play when we have functions that are difficult or impossible to integrate exactly. One of the most common numerical techniques is the Riemann sum.
Riemann sums divide the area under a curve into shapes (like rectangles) to approximate the total integral. Different types of Riemann sums, such as left endpoint, right endpoint, and midpoint, utilize different parts of the subintervals to estimate the area. Computer algebra systems often calculate Riemann sums, providing numeric approximations for integrals that are otherwise too complex to solve directly.

Subintervals

Subintervals are smaller sections into which a larger interval is divided. In numerical integration, like Riemann sums, subintervals are used to approximate the region under a curve for better accuracy.
In our exercise with \( \int_{0}^{1} \tan(x) \), the interval \([0,1]\) is divided into 10 subintervals, each \(0.1\) units wide. By using multiple subintervals, a Riemann sum method provides a finer approximation of the integral. The more subintervals you use, the closer you get to the exact answer, but this also increases the computation effort.

Trigonometric Functions

Trigonometric functions, like \( \tan(x) \), are fundamental in mathematics and widely used in science and engineering. These functions relate angles of triangles to the lengths of their sides. The tangent function, specifically, is the ratio of the opposite to the adjacent side in a right triangle.
When analyzing \( \tan(x) \) from \( 0 \) to \( 1 \), it's important to note that \( \tan(x) \) can have steep inclines. This makes numerical methods, such as Riemann sums, quite valuable due to their ability to break down the evaluation into manageable segments, each handled individually. Understanding these functions, their properties, and how they behave graphically helps immensely in solving and approximating integrals involving them.