∫tan x limit “a” to ”b” by limit as sum method

and

∫cos x limit “a” to ”b” by limit as sum method

To evaluate the integrals of \(\tan x\) and \(\cos x\) from \(a\) to \(b\) using the limit of a sum method, we will break down the process step by step. This method is based on the definition of the definite integral as the limit of Riemann sums. Let's dive into each integral separately.

Evaluating the Integral of \(\tan x\)

We start with the integral of \(\tan x\) from \(a\) to \(b\). The first step is to express the integral as a limit of Riemann sums.

Step 1: Setting Up the Riemann Sum

We divide the interval \([a, b]\) into \(n\) equal subintervals, each of width \(\Delta x = \frac{b - a}{n}\). The endpoints of these subintervals can be represented as:

• x_i = a + iΔx for i = 0, 1, 2, ..., n

Now, we can express the Riemann sum for \(\tan x\) as:

Riemann Sum = \(\sum_{i=1}^{n} \tan(x_i^*) \Delta x\), where \(x_i^*\) is a sample point in each subinterval. A common choice is to take \(x_i^* = x_i\).

Step 2: Writing the Limit

The integral can then be expressed as:

\(\int_a^b \tan x \, dx = \lim_{n \to \infty} \sum_{i=1}^{n} \tan(a + i \Delta x) \Delta x\)

Step 3: Evaluating the Limit

As \(n\) approaches infinity, \(\Delta x\) approaches zero, and we can evaluate the limit. However, the integral of \(\tan x\) does not have a simple antiderivative, so we typically use known results or numerical methods for evaluation. The antiderivative of \(\tan x\) is \(-\ln|\cos x|\), so we can evaluate:

\(\int_a^b \tan x \, dx = -\ln|\cos b| + \ln|\cos a| = \ln\left(\frac{|\cos a|}{|\cos b|}\right)\)

Calculating the Integral of \(\cos x\)

Next, let’s evaluate the integral of \(\cos x\) from \(a\) to \(b\) using the same limit of sums approach.

Step 1: Riemann Sum for \(\cos x\)

Similar to before, we divide the interval \([a, b]\) into \(n\) equal parts. The Riemann sum for \(\cos x\) is given by:

Riemann Sum = \(\sum_{i=1}^{n} \cos(x_i^*) \Delta x\)

Step 2: Expressing the Integral

Thus, we can write:

\(\int_a^b \cos x \, dx = \lim_{n \to \infty} \sum_{i=1}^{n} \cos(a + i \Delta x) \Delta x\)

Step 3: Finding the Limit

The antiderivative of \(\cos x\) is \(\sin x\). Therefore, we can evaluate the definite integral as follows:

\(\int_a^b \cos x \, dx = \sin b - \sin a\)

Summary of Results

In summary, we have:

• For \(\int_a^b \tan x \, dx\): \(\ln\left(\frac{|\cos a|}{|\cos b|}\right)\)
• For \(\int_a^b \cos x \, dx\): \(\sin b - \sin a\)

This method of using Riemann sums provides a solid foundation for understanding the concept of integration, and while the calculations can become complex, the underlying principles remain consistent across different functions.