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Chapter 2: Problem 128

How is integration related to derivation?

Short Answer

Expert verified
Question: Explain the relationship between integration and differentiation, and provide an example. Answer: Integration and differentiation are inverse operations, meaning that one process essentially undoes the other. They are two fundamental concepts in calculus; differentiation is the process of finding the derivative or rate of change of a function, and integration is the process of finding the antiderivative or integral of a function. If you have the derivative of a function (f'(x)), and you integrate it, you will get back the original function (f(x)) plus a constant (C). For example, if f(x) = 2x^2, its derivative, f'(_x), equals 4x. Integrating f'(x) gives back the original function, ∫4x dx = 2x^2 + C.

Step by step solution

01

Define differentiation

Differentiation is the process of finding the derivative of a function. The derivative represents the rate of change or slope of a function at a given point. In mathematical terms, if a function is given by f(x), then its derivative is denoted by f'(x) or df/dx, and it measures the change in y (output) with respect to x (input).
02

Define integration

Integration is the opposite process of differentiation. It is the process of finding the antiderivative or the integral of a given function. The integral represents the net area under the curve of a function. In mathematical terms, suppose F(x) is the antiderivative of f(x), then the integral of f(x) is written as ∫f(x)dx = F(x) + C, where C is the constant of integration.
03

Explain the relationship between differentiation and integration

Integration and differentiation are inverse operations, meaning that one process essentially undoes the other. If you have the derivative of a function (f'(x)), and you integrate it, you will get back the original function (f(x)) plus a constant (C).
04

Provide an example

Let's consider a simple example: - Let f(x) = 2x^2 - Find the derivative, f'(x), by applying the power rule: f'(x) = 4x (using the power rule where the derivative of ax^n = nax^(n-1)) - Now, integrate f'(x) with respect to x: ∫4x dx = 2x^2 + C (using the power rule of integration, ∫x^n dx = (x^(n+1))/(n+1) + C) As shown in this example, after differentiating the given function (f(x)), and then integrating its derivative (f'(x)), we obtained the original function (2x^2) plus a constant (C). This demonstrates the inverse relationship between differentiation and integration.

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Most popular questions from this chapter

Consider a \(1.5\)-m-high and \(0.6-\mathrm{m}\)-wide plate whose thickness is \(0.15 \mathrm{~m}\). One side of the plate is maintained at a constant temperature of \(500 \mathrm{~K}\) while the other side is maintained at \(350 \mathrm{~K}\). The thermal conductivity of the plate can be assumed to vary linearly in that temperature range as \(k(T)=\) \(k_{0}(1+\beta T)\) where \(k_{0}=18 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}\) and \(\beta=8.7 \times 10^{-4} \mathrm{~K}^{-1}\). Disregarding the edge effects and assuming steady onedimensional heat transfer, determine the rate of heat conduction through the plate. Answer: \(22.2 \mathrm{~kW}\)

The conduction equation boundary condition for an adiabatic surface with direction \(n\) being normal to the surface is (a) \(T=0\) (b) \(d T / d n=0\) (c) \(d^{2} T / d n^{2}=0\) (d) \(d^{3} T / d n^{3}=0\) (e) \(-k d T / d n=1\)

A spherical vessel has an inner radius \(r_{1}\) and an outer radius \(r_{2}\). The inner surface \(\left(r=r_{1}\right)\) of the vessel is subjected to a uniform heat flux \(\dot{q}_{1}\). The outer surface \(\left(r=r_{2}\right)\) is exposed to convection and radiation heat transfer in a surrounding temperature of \(T_{\infty}\). The emissivity and the convection heat transfer coefficient on the outer surface are \(\varepsilon\) and \(h\), respectively. Express the boundary conditions and the differential equation of this heat conduction problem during steady operation.

Consider a long rectangular bar of length \(a\) in the \(x-\) direction and width \(b\) in the \(y\)-direction that is initially at a uniform temperature of \(T_{i}\). The surfaces of the bar at \(x=0\) and \(y=0\) are insulated, while heat is lost from the other two surfaces by convection to the surrounding medium at temperature \(T_{\infty}\) with a heat transfer coefficient of \(h\). Assuming constant thermal conductivity and transient two-dimensional heat transfer with no heat generation, express the mathematical formulation (the differential equation and the boundary and initial conditions) of this heat conduction problem. Do not solve.

What is heat generation? Give some examples.
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