Question

Express the limit as a definite integral on the given interval [5pi,3pi].

Answer 1

To express a limit as a definite integral, we can consider the definition of the definite integral as the limit of a Riemann sum.

For example, if we want to express the limit

\[
\lim_{n \to \infty} \sum_{i=1}^{n} f(x_i^*) \Delta x
\]

for a function \( f(x) \) over the interval \([a, b]\), we can say that this limit represents the definite integral:

\[
\int_a^b f(x) \, dx
\]

In your case, you want to express a limit as a definite integral on the interval \([5\pi, 3\pi]\). Let's assume \( f(x) \) is a certain continuous function on this interval.

To properly set up the expression, we must have:

- Determine \( \Delta x = \frac{b - a}{n} = \frac{3\pi - 5\pi}{n} = \frac{-2\pi}{n} \)
- Choose points \( x_i^* \) in each subinterval, for instance, \( x_i^* = 5\pi + i \Delta x \).

Thus, the limit can be expressed as follows:

\[
\lim_{n \to \infty} \sum_{i=1}^{n} f\left(5\pi + i \left( \frac{-2\pi}{n} \right)\right) \left( \frac{-2\pi}{n} \right) = \int_{5\pi}^{3\pi} f(x) \, dx
\]

Therefore, the definite integral on the interval \([5\pi, 3\pi]\) is represented as:

\[
\int_{5\pi}^{3\pi} f(x) \, dx
\]

Note that since limits and integrations can be defined in either order (lower to upper bounds or vice versa), it’s often more standard to express integrals from \( a \) to \( b \) with \( a < b\). Thus, if \( a = 5\pi \) and \( b = 3\pi \), the integral can be rewritten as:

\[
-\int_{3\pi}^{5\pi} f(x) \, dx
\]

if \( f(x) \) is the function in question.