Answer to Question #269076 in Calculus for Amu Raj

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, , xy, , 10 .

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

some properties of indefinite integral.

Answer 1

Q1. Calculus is used extensively in mechanical engineering, such as calculating the surface area of complicated objects to estimate frictional forces, building a pump based on flow rate and head, and estimating the power delivered by a battery system.

Q2. Differentiation is the inverse of integration.... Differentiation is used to compute the function's speed as well as its immediate velocity. As it estimates the area under the curve, integration is used to compute the distance traveled by any function.

Q3.

$\int \sin \left(3x\right)dx\\ =\int \sin \left(u\right)\frac{1}{3}du\\ =\frac{1}{3}\left(-\cos \left(u\right)\right)\\ =\frac{1}{3}\left(-\cos \left(3x\right)\right)\\ =-\frac{1}{3}\cos \left(3x\right)+C$

$\int \:xydx\\ =y\frac{x^2}{2}\\ =\frac{yx^2}{2}+C$

$\int \:10dx\\ =10x+C$

Q4

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

These members can be obtained by shifting any one of the curves parallel to itself.

The above statement can be considered as some properties of indefinite integral

• ∫kf(x)dx=k∫f(x)dx , where k is any number.
• ∫−f(x)dx=−∫f(x)dx
• ∫f(x)±g(x)dx=∫f(x)dx±∫g(x)dx