Aim: To determine the value of 'c' in
an indefinite integral.
To do this, we usually will have to rely on extra information.
Here are some examples.
Find function y given that dy/dx = 3x^{2}
and the curve y passes through the point (2,13).
Solution:
y = ∫ 3x^{2} dx
y = x^{3}+ c ; now to find the y we have to use the given point
(2,13).
13 = 2^{3}+ c ------------------[substitute 2 into x and
13 into y ].
13 = 8 + c
c = 5.
y = x^{3}+ 5 is the answer.
A toy-plane is moving in a straight line with a velocity of 10 m/s at t=3
seconds. It's acceleration is modelled by a = 5 - 6t^{2}. Find the
equation of its velocity.
Solution:
v = ∫ 5 - 6t^{2} dt
v = 5t - 2t^{3}+ c ; now to find the y we have to use the given point
(3,10).
10 = 5(3) - 2(3)^{3}+ c
10 = 15 - 54 + c
c = 49.
v = 5t - 2t^{3}+ 49 is the answer.
The marginal cost of producing a table is dC/dq
= 6q -30 where q is the number of table produced. If no table is produced
then the producer will still have cost amounting to $250. Find the equation
C.
Solution: C = ∫ 6q-30^{} dq C = (6/2)q^{2}- 30q + c ; now to find the
c we have to use the given fact that C=250 when
q=0.
250 = 3(0)^{2}- 30(0) + c
c = 250 C = 3q^{2}- 30q + 250 is the answer.
[In economics, the c=$250 is called the fixed costs.]