Calculus 1 Lecture Notes, Section 4.1



Integration: The Other Half of Calculus

Integration Overview

1. Functionally speaking, integration is the inverse process of differentiation:

|Analogy |

|Division is the inverse process of multiplication. |Integration is the inverse process of differentiation. |

|The process is called dividing. |The process is called integrating. |

|Mathematical notation for division: |Mathematical notation for integration: |

|[pic] |[pic] |

|“ten divided by 2 equals x” |“the integral of the function ax2 with respect to the variable x |

| |equals F(x)” |

|means find a number x such that | |

|[pic] |means find a function F(x) such that |

| |[pic] |

|x = 5, since [pic] |[pic], since [pic] |

|x is called the quotient |F(x) is called the antiderivative |

|[pic] is called a fraction |[pic] is called an (indefinite) integral |

2. Geometrically speaking, integration is finding the area bounded by a function, the x axis, and two vertical lines at fixed x values:

|[pic] |[pic] |

|The symbol [pic]stands for Sum, because the shaded area is calculated by |[pic]is called a definite integral |

|partitioning the region into an infinite number of little vertical | |

|rectangles and then summing up their areas. | |

Section 4.1: Antiderivatives

1. An antiderivative of a function f(x) is any function F(x) such that [pic].

2. One of the most common functions to integrate is the power function. To do this, we “run the power rule in reverse”:

|Power Rule for taking derivatives: |Power Rule “in reverse” for taking antiderivatives: |

|[pic] |[pic] |

|Example: find the derivative of [pic] |Example: find the antiderivative of [pic] |

|Multiply by the exponent. |Add one to the exponent. |

|[pic] |[pic] |

|Subtract one from the exponent. |Divide by the new exponent. |

|[pic] |[pic] |

| |Add a constant of integration. |

| |[pic] |

a. Practice: find three antiderivatives of f(x) = ½x4.

3. Notice that if F(x) is an antiderivative of f(x), then F(x) + c (where c is any constant) is also an antiderivative of f(x) (this is theorem 1.1).

4. The “indefinite integral of f(x) with respect to x” is written in mathematical notation as: [pic], where F(x) is an antiderivative of f(x) (this is definition 1.1).

a. c is called the constant of integration

b. f(x) is called the integrand

c. dx identifies the variable x as the variable of integration

d. Practice: Evaluate the indefinite integral [pic]

Note: to check your work graphically, you can use Winplot (or your graphing calculator) to graph the function and its antiderivative. Then use Winplot to graph the derivative of the antiderivative. That curve should lie exactly on top of the original function.

[pic]

e. Practice: Evaluate the indefinite integral [pic]

f. Practice: Evaluate the indefinite integral [pic]

5. We can compute many integrals by “rearranging” derivatives that we already know:

|Power Rule |[pic] |because |[pic] |

|(Thm. 1.2) |(provided r ( -1) | | |

|Trig |[pic] |because |[pic] |

|Functions | | | |

| |[pic] |because |[pic] |

| |[pic] |because |[pic] |

| |[pic] |because |[pic] |

| |[pic] |because |[pic] |

| |[pic] |because |[pic] |

|Exponential |[pic] |because |[pic] |

|Function |[pic] | |And |

| | | |[pic] |

|Reciprocal |[pic] |because |[pic] for x>0 |

|Function |(provided x ( 0) | |And |

|(Corollary 1.1) | | |[pic] |

| | | |for x>0 |

|Inverse Trig |[pic] |because |[pic] |

|Functions | | |And |

| | | |[pic] |

| |[pic] |because |[pic] |

| | | |And |

| | | |[pic] |

| |[pic] |because |[pic] |

|Inverse Hyperbolic |[pic] |because |[pic] |

|Functions | | | |

| |[pic] |because |[pic] |

| |[pic] |because |[pic] |

| |[pic] |because |[pic] |

|Hyperbolic Functions |[pic] |because |[pic] |

| |[pic] |because |[pic] |

| |[pic] |because |[pic] |

6. We also can find more complicated integrals by making use of properties of derivatives and the Chain Rule:

|Functions |[pic] |because |[pic] |

|multiplied by | | | |

|constants and | | | |

|summed | | | |

|(Thm. 1.3) | | | |

|Argument of |[pic] |because |[pic] |

|function |(for a ( 0) | | |

|multiplied by | | | |

|constant | | | |

|Quotient of a |[pic] |because |[pic] |

|derivative and a |(provided f(x) ( 0) | | |

|function | | | |

|(Corollary 1.2) | | | |

| |[pic] |because |[pic] |

|Product of a |[pic] |because |[pic] |

|function and its | | | |

|derivative | | | |

a. Practice: [pic]

b. Practice: [pic]

c. Practice: [pic]

d. Practice: [pic]

e. Practice: [pic]

f. Practice: [pic]

g. Practice: [pic]

h. Practice: [pic]

i. Find f(x) satisfying f ((x) = 4cos x if f(0) = 3.

j. Find all f(x) satisfying f (((x) = 4cos x + 3x.

k. Determine the position function s(t) if the velocity function is v(t) = 3e-t – 2 and the initial position is s(0) = 1.

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