Answer to Question #269515 in Calculus for khan

Question

Q.1: What are the applications of Calculus in engineering?

Q.1: Define differentiation and integration with example. What are the differences between them?

Q.3: Integrate the following functions with respect to x:

Sin3x, x^6 , xy, e^5x , 10 .

Q.4: Describe geometrical meaning of indefinite integral. Write down

some properties of indefinite integral.

Answer 1

Engineering

Many aspects of civil engineering require calculus. Firstly, derivation of the basic fluid mechanics equations requires calculus. For example, all hydraulic analysis programs, which aid in the design of storm drain and open channel systems, use calculus numerical methods to obtain the results. In hydrology, volume is calculated as the area under the curve of a plot of flow versus time and is accomplished using calculus.

Structural Engineering

In structural engineering, calculus is used to determine the forces in complex configurations of structural elements. Structural analysis relating to seismic design requires calculus. In a soil structure context, calculations of bearing capacity and shear strength of soil are done using calculus, as is the determination of lateral earth pressure and slope stability in complex situations.

Mechanical Engineering

Many examples of the use of calculus are found in mechanical engineering, such as computing the surface area of complex objects to determine frictional forces, designing a pump according to flow rate and head, and calculating the power provided by a battery system. Newton's law of cooling is a governing differential equation in HVAC design that requires integration to solve.

Aerospace Engineering

Numerous examples of the use of calculus can be found in aerospace engineering. Thrust over time calculated using the ideal rocket equation is an application of calculus. Analysis of rockets that function in stages also requires calculus, as does gravitational modeling over time and space. Almost all physics models, especially those of astronomy and complex systems, use some form of calculus.

Differentiation is the process of finding the derivative of a function while Integration is the process of finding the integral of a function.

An example of differentiation is

 $y = x^2\\ \implies \frac{dy}{dx}=2x$

An example of integration is

$\frac{dy}{dx}= 2x\\ \implies \int \frac{dy}{dx} = \int 2x\\ =y = x^2 +c$

$\displaystyle \int \sin3xdx = -\frac{\cos3x}{3}+c\\ \int x^6dx= \frac{x^7}{7}+c\\ \int xy dx = \frac{x^2y}{2}+c\\ \int e^{5x}= \frac{e^{5x}}{5} +c\\ \int 10dx = 10x+c$

Consider a curve which is above the x-axis. It is a continuous function on the interval [a, b] where all the values are positive. The area between the curve and the x-axis defines the definite integral

$\displaystyle \int^b_a f(x)dx = F(b)-F(a)$

of any continuous functions. In the above formulae a and b are the limits.

Properties Of Definite Integral

$\displaystyle \int ^a_bf(x)dx= \int ^a_bf(t)dt\\ -\int ^a_bf(x)dx= \int ^b_af(x)dx\\ \int ^b_bf(x)dx= \int ^b_bf(a+b-x)dx$