a) Find the instantaneous rate of change of V b) Find the rate of change of the volume if the length of one side is 5 cm. c) Find the surface area (A) of the metal cube. d) Find the instantaneous rate of change of the surface area e) Find the rate of change of the surface area if the length of one side is 5 cm INTEGRATION Outcomes for this section: By the end of this section, you should be able to: Understand the concept of integration as a summation function (definite integral) and as converse of differentiation (indefinite integral). Apply standard forms of integrals as a converse of differentiation. Integrate the following functions: with n ; with kxn, and with > 0 ; kanx a k , a Integrate polynomials consisting of terms of the above forms. Apply integration to determine the magnitude of an area included by a curve and the or by a curve, the and the ordinates = and = where . x-axis x-axis x a x b a , b Introduction The process of integration reverses the process of differentiation. In differentiation, if f(x) = 2x2, then (x) = 4x. Thus the integral of 4x is 2x2. We can represent this process pictorially as follows: The situation gets a bit more complicated, because there are an infinite number of functions we can differentiate to give 4x. Here are some of those functions: = 2x2 + 7 ; = 2x2 8 ; = 2x2 + . f(x) g(x) h(x) – Activity 1 Write down at least five other functions whose derivative is 4x. You would have noticed that all the functions have the same derivative of 4x, because when we differentiate the constant term we obtain zero. Hence, when we the process, we reverse have no idea what the original constant term might have been. So we include in our response an unknown constant, called the arbitrary constant of integration. The integral of 4x then c, is 2x2 + c.
a) Find the instantaneous rate of change of V b) Find the rate of change of the volume if the length of one side is 5 cm. c) Find the surface area (A) of the metal cube. d) Find the instantaneous rate of change of the surface area e) Find the rate of change of the surface area if the length of one side is 5 cm INTEGRATION Outcomes for this section: By the end of this section, you should be able to: Understand the concept of integration as a summation function (definite integral) and as converse of differentiation (indefinite integral). Apply standard forms of integrals as a converse of differentiation. Integrate the following functions: kxn, with n ; with and kanx with a > 0 ; k , a Integrate polynomials consisting of terms of the above forms. Apply integration to determine the magnitude of an area included by a curve and the x-axis or by a curve, the x-axis and the ordinates x = a and x = b where a , b . Introduction The process of integration reverses the process of differentiation. In differentiation, if f(x) = 2x2, then (x) = 4x. Thus the integral of 4x is 2x2. We can represent this process pictorially as follows: The situation gets a bit more complicated, because there are an infinite number of functions we can differentiate to give 4x. Here are some of those functions: f(x) = 2x2 + 7 ; g(x) = 2x2 – 8 ; h(x) = 2x2 + . Activity 1 Write down at least five other functions whose derivative is 4x. You would have noticed that all the functions have the same derivative of 4x, because when we differentiate the constant term we obtain zero. Hence, when we reverse the process, we have no idea what the original constant term might have been. So we include in our response an unknown constant, c, called the arbitrary constant of integration. The integral of 4x then is 2x2 + c.