91Ó°ÊÓ

A definite integral is an integral with specific upper and lower bounds, which computes the net area under the curve of a function between these two points.
Consider the integral expression:

• \(\int_{a}^{b} f(x) \, dx \)

Here, \(a\) and \(b\) are the constants defining the limits.
This integral results in a real number, representing the accumulated quantities, such as areas or volumes, depending upon the equation provided under the curve.
It’s different from an indefinite integral, which has an arbitrary constant of integration, usually denoted as \(C\).
In this specific exercise, since the integral is definite, meaning the upper limit \(10\) and the lower limit \(3\sqrt{2}\) are constants, any derivative taken with respect to a variable not included in the limits (here, \(x\)) results in zero. This is because changing \(x\) doesn't affect the integral.

Leibniz rule is a handy tool in calculus when differentiating an integral with variable limits.
For a more complex integral with limits as functions of \(x\), Leibniz’s rule helps in getting the derivative in situations where the limits affect the function under the integral sign:

• \(\frac{d}{dx} \int_{u(x)}^{v(x)} f(t) \, dt = f(v(x)) \cdot v'(x) - f(u(x)) \cdot u'(x) + \int_{u(x)}^{v(x)} \frac{\partial}{\partial x}f(t, x) \, dt\)

In this exercise, since both the upper and lower bounds of the integral are constants \(10\) and \(3\sqrt{2}\), they don't depend on \(x\).
Thus, using Leibniz's rule in its simplest form, the derivative of this definite integral with respect to \(x\) is simply \(0\).
The presence of fixed limits implies that their changes don't impact the value of the integral as it isn't dependent on \(x\).

In calculus, a derivative represents the rate of change of a function with respect to a variable.
It's a fundamental tool for understanding how functions vary, similar to how speed indicates how distance varies over time.
Mathematically, the derivative of a function \(f(x)\) with respect to \(x\) is symbolized as \(\frac{df}{dx}\).
In the context of this exercise, although we have an integral which usually involves a variable in the integrand being derived, our derivative here is straightforward:

• The task is to find \(\frac{dy}{dx}\) where \(y = \int_{3 \sqrt{2}}^{10} \ln(2 + p^2) \, dp \), and since \(y\) is entirely independent from \(x\), we see \(\frac{dy}{dx} = 0\).

This highlights how derivatives work in cases where the independent variable doesn't interact with the defined integral, underscoring the relationships separate from dependency on \(x\).
It's a great illustration of how constants and unrelated variables in calculus can streamline computations.