Question

Practical situation(s) where process of integration is involved is/are
(a)We can determine the speed of the object if we know its instantaneous velocity
(b)We can determine the maximum velocity of the object if we know its instantaneous velocity
(c)We can determine the position of the object at any instant if we know its instantaneous velocity
(d)All of the above

Answer 1

Hint: Use the fact that we can get the exact equation of any function by integrating the partial derivative of a function at any instant.

Complete step-by-step answer:
 We will solve this question by checking each and every option and then deciding if it’s correct.
So, let’s begin by examining option (a).
The statement says that we can determine the speed of the object if we know its instantaneous velocity.
We know that the speed of any object is a scalar quantity. It is indeed the velocity of the object without describing the direction in which the object is moving.
Thus, the instantaneous speed of an object is just the instantaneous velocity of the object without the direction. Thus, we don’t need integration to find the speed of an object.
Option (a) is incorrect.
We will now examine option (b). It says that we can determine the maximum velocity of the object if we know its instantaneous velocity.
Instantaneous velocity of an object is given by the function \[\dfrac{dv}{dt}\].
We can find the maximum velocity of an object by equating \[\dfrac{dv}{dt}\] it to zero and solving the equation to find the maximum value of velocity.
Thus, we don’t need integration to find the maximum velocity. Option (b) is incorrect.
We will now examine option (c). It states that we can determine the position of the object at any instant if we know its instantaneous velocity.
We know that the relation between position of object and instantaneous velocity is given by \[v=\dfrac{dx}{dt}\]
We can rewrite this equation as \[vdt=dx\]. Thus, we can find the position of an object by integrating the instantaneous velocity with respect to time.
Thus, option (c) is correct.
Option (d) is incorrect as option (a) and (b) are incorrect.
Hence, the correct answer is (c).
Answer is Option (c)
Note: We need to carefully use the relations between instantaneous velocity and position of an object to check all the options and decide which situation uses integration to solve the problem. Even if the sentence is correct, it doesn’t mean that we are using integration to solve the problem.