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

 Q.1

i) Computing the surface area of complex objects to determine frictional forces

ii) Designing a pump according to flow rate and head

iii) Calculating the power provided by a battery system

Q.2

Differentiation is a process of finding the instantaneous rate of change in function based on one of its variables. For Example, differentiate the function $y=x^2$ .Then our solution is ${dy\over dx}=2x$

Integration is a method to find definite and indefinite integrals. The integration of a function f(x) is given by F(x)

For example, integrate the function $2x$ .Then our solution is $\int 2xdx={2x^{1+1}\over {1+1}}=x^2$

The differences between differentiation and Integration is that Integration is the reverse of differentiation

Q.3:

i) $\int sin3xdx$

Let $u=3x \implies {du\over dx}=3 \implies dx={1\over 3}du$

Now, Substitute

$={1\over 3}\int sin(u)du$

$={1\over 3}[-cos(u)]$

$={1\over 3}[-cos(3x)]$

$\therefore \int sin3xdx= {-cos(3x)\over 3}+C$

ii) $\int x^6dx={x^{6+1}\over 6+1}={1\over 7}x^7+C$

iii) $\int xydx =y\int xdx=y{x^{1+1}\over 1+1}={1\over 2}yx^2+C$

iv) $\int e^{5x}dx$

Let $u=5x \implies {du\over dx}=5 \implies dx= {du\over 5}$

Substitute

$\int e^{u}{du\over 5}={e^u\over 5}+C={e^{5x}\over 5}+C$

v) $\int 10dx=10\int dx=10x+C$

Q.4:

$\int f(x) dx =F(x)+C$ represents a family of curves and different values of C corresponds to different members of this family.

Properties of indefinite integral

1) $\int cf(x)dx=c\int f(x)dx$

From the indefinite integral, we can take out the multiplicative constants.

2) $\int -f(x)dx=\int f(x)dx$

Due to the negative function, the indefinite integral is also negative.

3) $\int [f(x) \pm g(x)]dx=\int f(x)dx\pm \int g(x)dx$

It shows the sum as well as the difference of the integral of the functions as the sum or the difference of their individual integral.