Integration Chapter 15

Integration 15.1 Antiderivatives 15.3 Area and Definite Integral 15.4 The Fundamental Theorem of Calculus

Differential and Integral Calculus Differential calculus determines the rate of change at which a quantity is changing. Example: f(x) = x5; find f ’(x). Integral calculus finds the total change in quantity where the rate of change is known. Example: f ’(x) = 5x4; find f(x).

15.1 Antiderivatives In previous chapters, functions provided information about a total amount q. Cost C(q), revenue R(q), profit P(q),... Derivatives of these functions provide information about the rate of change and extrema. If the rate of change is known, the antiderivative is calculated to determine the function and the total quantity. ANTIDERIVATIVE If the derivative of F(x) = f(x), then F(x) is an antiderivative of f(x). The process of finding antiderivatives is called antidifferentiation or indefinite integration.

Antiderivatives Examples If F(x) = 10x, then F’(x) = 10, so F(x) = 10x is an antiderivative of f(x) = 10. For F(x) = x2, F’(x) = 2x, making F(x) = x2 an antiderivative of f(x) = 2x. Find an antiderivative of f(x) = 5x4 f(x) = F‘(x) = 5x5 – 1 Therefore, the antiderivative of f(x) = x5 Verify that is an antiderivative of f(x) = x2 + 5 F(x) is an antiderivative of f(x) if, and only if, F’(x) = f(x)

Antiderivatives Therefore, is an antiderivative of f(x) = x2 + 5

The General Antiderivative of a Function A function has more than one antiderivative. One antiderivative of the function f(x) = 3x2 is F(x) = x3, since F’(x) = 3x2 = f(x) but so are x3 + 10, and x3 – 5, and x3 + , since This means that there is a family or class of functions that are antiderivatives of 3x2. If F(x) and G(x) are both antiderivatives of a function f(x) on an interval, than there is a constant C such that F(x) – G(x) = C Two antiderivatives of a function can differ only by a constant. The arbitrary real number C is called an integration constant.

F(x), G(x), and H(x) are antiderivatives of 3x2 G’(x) = g(x) = 3x2 F’(x) = f(x) = 3x2 H’(x) = h(x) = 3x2 F(x), G(x), and H(x) are antiderivatives of 3x2

F(x), G(x), and H(x) are antiderivatives of 3x2 C = 10 F(x), G(x), and H(x) are antiderivatives of 3x2 C = -5 Each graph can be obtained from another by a vertical shift of |C| units. The “family” of antiderivatives of f(x) is represented by F(x) + C.

Antiderivatives The family of all antiderivatives of the function f is indicated by The symbol ∫ is the integral sign, f(x) is the integrand, and ∫ f(x) dx is called an indefinite integral, the most general antiderivative of f. INDEFINITE INTEGRAL If F’(x) = f(x), then for any real number C

Antiderivatives ∫ 2ax dx = ∫ a(2x)dx = ax2 + C Using this notation, ∫ 2x dx = x2 + C the dx indicates that ∫ f(x) dx is the “integral of f(x) with respect to x”. In ∫ 2ax dx, the dx indicates that a is a constant and x is the variable of the function. Therefore, ∫ 2ax dx = ∫ a(2x)dx = ax2 + C And ∫ 2ax da = ∫ x(2a) da = xa2 + C

Rules For Finding Antiderivatives CONSTANT MULTIPLE RULE For constant k The indefinite integral of a constant is the constant itself.

Rules For Finding Antiderivatives POWER RULE For any real number n  -1, To find the indefinite integral of a variable x raised to an exponent n, increase the exponent by 1 and divide by the new exponent, n + 1.

Rules For Finding Antiderivatives SUM OR DIFFERENCE RULE The indefinite integral of the sum or difference of two functions f(x) and g(x) with respect to x, is the sum or difference of the indefinite integrals of the functions. (Constant rule) (Power rule)

Example The marginal profit of a small fast-food stand is given by P’(x) = 2x + 20 where x is the sales volume in thousands of hamburgers. The “profit” is –$50 when no hamburgers are produced. Find the profit function. (Used k instead of C to avoid confusion with “Cost”)

Rules For Finding Antiderivatives INDEFINITE INTEGRALS OF EXPONENTIAL FUNCTIONS

Indefinite Integrals Of Exponential Functions Examples

Rules For Finding Antiderivatives POWER RULE FOR n = -1

Now You Try Find the cost function. Constant Multiple Rule Sum Rule Power Rule

Now You Try Find the cost function. Constant Multiple Rule

15.3 Area and the Definite Integral Definite integral: Can be defined as the area of the region under the graph of a function f on the interval [a, b]

f(x) Area under the graph of x

Area and the Definite Integral Definite integral: Can be defined as the area of the region under the graph of a function f on the interval [a, b] Based upon the concept that a geometric figure is a sum of other figures.

f(x) x Sum of the areas of the rectangles provides a rough approximation of the area under the curve.

f(x) x Increasing the number of rectangles results in a closer approximation of the area under the curve.

f(x) x Increasing the number of rectangles results in a closer approximation of the area under the curve.

Area and the Definite Integral Definite integral: Can be defined as the area of the region under the graph of a function f on the interval [a, b] Based upon the concept that a geometric figure is the sum of other figures. The area under the curve can be determined by summing the areas of n rectangles of equal width. If the interval is from x = 0 to x = 2, then the width of each rectangle equals and height determined by f(x)

f(x) f(0) = 2 x 1

Area Under the Curve n Area 2 3.732 4 3.496 8 3.340 10 3.305 20 3.229 50 3.178 100 3.160 500 3.146 n Area 2000 3.143 8000 3.142 32,000 3.1417 128,000 3.1416 512,000 3.14159

Definite Integral Calculating the area under the curve of a function on the interval [a, b] Divide the area between x = a, and x = b into n intervals Let f(xi) = the height if the ith rectangle (The ith rectangle is an arbitrary rectangle) ∆x = the width of each of the rectangles Then the area of the ith rectangle = f(xi) • ∆x and the total area under the curve is approximated by the sum of the areas of all n rectangles, or

Definite Integral THE DEFINITE INTEGRAL If f is defined by the interval [a, b], the definite integral of f from a to b is given by provided the limit exists, where ∆x = (b – a)/n and xi is any value of x in the ith interval. The definite integral of f(x) on [a, b] is written as and approximated by

15.4 The Fundamental Theorem of Calculus As seen in section 15.3, gives the area between the graph of f(x) and the x-axis, from x = a to x = b. This area can be determined using the antiderivatives discussed in section 15.1 If f(x) gives the rate of change of F(x), then F(x) is an antiderivative of f(x) and gives the total change of F(x) as x changes from a to b which can also be expressed as F(b) – F(a)

The Fundamental Theorem of Calculus A real estate agent wants to evaluate an unimproved parcel of land that is 100 feet wide and is bounded by streets on three sides and by a stream on the fourth side. The agent determines that if a coordinate system is set up as shown in the next slide, the stream can be described by the curve y = x3 + 1, where x and y are measured in hundreds of feet.

y (100 ft) Stream 1 x (100ft) 1

The Fundamental Theorem of Calculus A real estate agent wants to evaluate an unimproved parcel of land that is 100 feet wide and is bounded by streets on three sides and by a stream on the fourth side. The agent determines that if a coordinate system is set up as shown in the next slide, the stream can be described by the curve y = x3 + 1, where x and y are measured in hundreds of feet. If the area of the parcel is A square feet and the agent estimates the land is worth $12 per square foot, then the total value of the parcel is 12A dollars. How can the agent find the area and hence the total value of the parcel?

The Fundamental Theorem of Calculus Allows us to compute definite integrals and thus find area and other quantities by using the indefinite integration methods. THE FUNDAMENTAL THEOREM OF CALCULUS If the function f(x) is continuous on the interval [a, b], then where F(x) is any antiderivative of f(x) on [a, b] When applying the Fundamental Theorem, use the notation

The Fundamental Theorem of Calculus Find the area of the parcel of land described in the introduction to this section. That is, the area under the curve y = x3 + 1 on the interval [0, 1] The area of the parcel is given by the definite integral Since an antiderivative of f(x) = x3 + 1 is the fundamental theorem of calculus tells us that

The Fundamental Theorem of Calculus Because x and y are measured in hundreds of feet, the total area is Since the land is worth $12 per square foot, the total value of the parcel is

Another example Find the area below the graph of f(t) = 4t3 and above the x axis between t = 1 and t = 2.

Another example Find the area below the graph of f(t) = 4t3 and above the x axis between t = 1 and t = 2. Solution:

Properties of Definite Integrals for any real constant k (constant multiple of a function) (sum or difference of a function) for any real number c that lies between a and b

Example (sum or difference of a function) (constant multiple of a function) (power rule)

Region lies below the x-axis The area is given by

By the Fundamental Theorem,

Example A worker new to a job will improve his efficiency with time so that it takes him fewer hours to produce an item with each day on the job, up to a certain point. Suppose the rate of change of the number of hours it takes a worker to produce the xth item is given by H’(x) = 20 – 2x What is the total number of hours required to produce the first 5 items? What is the total number of hours required to produce the first 10 items?

Example 75 hours are required to produce the first 5 items.

y (hours) 75 x (items) 5 [0, 5]

Example 100 hours are required to produce the first 10 items. or

y (items) x (hours) [0, 10]

Now You Try. Given: . Find the area between the x axis and f(x) from x = 0 and x = 2.

Applications of Integrals Consumers’ and Producers’ Surplus Equilibrium price – The price at which quantity demanded equals quantity supplied. Some buyers are willing to pay more. Consumers’ Surplus – The total of the differences between the equilibrium price and the higher prices consumers would be willing to pay.

Applications of Integrals Consumers’ Surplus Equilibrium Price Producers’ Surplus

Applications of Integrals Equilibrium Price

Applications of Integrals Consumers’ Surplus Equilibrium Price

Applications of Integrals Consumers’ Surplus If D(q) is a demand function with equilibrium price p0 and equilibrium quantity q0, then p0 q0

Applications of Integrals Consumers’ and Producers’ Surplus Equilibrium price – The price at which quantity demanded equals quantity supplied. Some buyers are willing to pay more. Consumers’ Surplus – The total of the differences between the equilibrium price and the higher prices consumers would be willing to pay. Producers’ Surplus – The total of the differences between the equilibrium price and the lower prices sellers would be willing to charge.

Applications of Integrals Equilibrium Price

Applications of Integrals Equilibrium Price Producers’ Surplus

Applications of Integrals Producers’ Surplus If S(q) is a supply function with equilibrium price p0 and equilibrium quantity q0, then p0 q0

Applications of Integrals Finding the consumers’ and producers’ surplus Calculate equilibrium quantity (q0) and price (p0).

Applications of Integrals

Applications of Integrals

Applications of Integrals $4500 p0 = $375 $3375 q0 = 15

Now You Try Find the consumers’ and producers’ for an item having the following supply and demand functions

 Chapter 15 End