The Indefinite Integral
We begin with a question.
Question: List two functions F(x) such that F'(x) = x
Answer: 1/2 x2 and 1/2 x2 + 3
We can see that if
F'(x) = x
F(x) = 1/2 x2 + C
for some constant C.
We call F(x) the antiderivative or integral of f(x) and write
In general, if
F'(x) = f(x)
then we write
From the derivative formula
We get the integral formula
Just like with derivatives, to find an antiderivative of a sum or difference, we can take the antiderivative of each term. Also like derivatives, the antiderivative of the product or quotient is not easily found.
Which of these has an easy to find antiderivative
A. 8x3 - 6x
We can find the antiderivative of part A. easily, by finding the antiderivative of 8x3 and 6x separately. The antiderivative is
8(1/4 x4) - 6(1/2 x2) + C = 2x4 - 3x2 + C
B. This one, on the other hand, is a quotient. We do not have a way of finding its antiderivative.
Find the following integrals:
We have seen that an integral produces a whole family of solutions parameterized by C. In most applications, we are given an initial or other condition and hence find the value of C. The antiderivative with known C is called a particular solution.
Find a solution to
F'(x) = 4x - 3
F(1) = 2
We first find an antiderivative:
F(x) = 2x2 - 3x + C
Now plug in 1 for x and 2 for F to get:
2 = 2(1)2 - 3(1) + C = -1 + C
So that C = 3. The particular solution is
F(x) = 2x2 - 3x + 3.
Since the acceleration of gravity is a constant a = 32, we can derive the physics equations.
Suppose that we kick a football with an initial upward velocity of 100 feet per second how long will it take to hit the ground?
v(t) = -32 dt = -32t + C
v(0) = 100 = C
s(t) = (-32t + 100)dt = -16t2 + 100t + C
s(0) = 0 = C
s(t) = -16t2 + 100t = t(-16t + 100)
s(t) = 0 when -16t + 100 = 0
t = 100/16 = 6.25
It will take 6.25 seconds to hit the ground.
Suppose the marginal revenue for a ski resort is
M = 50 - 0.01 x
And suppose that at $50 per ticket, the ski resort will have 2,000 skiers.
Find the demand equation.
Since the marginal revenue is the derivative of the revenue, the revenue is the antiderivative of the marginal revenue.
R = (50 - 0.01x)dx = 50x - 0.005 x2 + C
The revenue is equal to the price times the quantity. That is
50x - 0.005 x2 + C = px
Now find C by noting that when p = 50, x = 2,000.
50(2,000) - 0.0005 (2,000)2 + C = (50)(2,000)
80,000 + C = 1,000,000
C = 920,000
Substituting the C into our equation and dividing by x gives the demand equation
p = 50 - 0.005 x + 920,000/x