Integration and Antiderivatives
Integration/Antiderivative
First let’s talk about what the
integral means!
Can you list some interpretations of
the definite integral?
Here’s a few facts
:
1. If f(x) > 0, then
returns the
numerical value of the area between
f(x) and the x-axis (area “under” the curve)
2.
= F(b) – F(a) where F(x) is
any anti-derivative of f(x).
(Fundamental Theorem of Calculus)
3. Basically gives the total cumulative
change in f(x) over the interval [a,b]
What is a Riemann Sum?
Hint: Here’s a picture!
A Riemann sum is the area of n rectangles used to
approximate the definite integral.
= area of n rectangles
As n approaches infinity…
and
So the definite integral sums
infinitely many infinitely thin
rectangles! (
Calculus trivia: as n (number of
rectangles) goes to the summation sign becomes the
integral sign and x becomes dx)
The indefinite integral
= ?
Well…hard to write; easy to say
The indefinite integral equals the
general antiderivative…
= F(x) + C
Where F’(x) = f(x)
= ax + C
=
+ C
= - cos x + C
Don’t forget we are going backwards!
So if the derivative was positive, the
anti-derivative is negative.
= sin x + C
= ln |x| +C
You need the absolute value in case x<0
where n > 1
Hint:
1/x
n
= x
-n
sooooooo…….
the answer is:
+ C
You didn’t say ln(x
n
) did ya??
= e
x
+ c
Easiest anti-derivative in the universe, eh?
Examples:
Exploring the concepts of definite and indefinite integrals, Riemann sums, and antiderivatives in calculus. Learn about interpreting the definite integral, Riemann sums as rectangles approximating integrals, and finding general antiderivatives. Discover various formulas for finding antiderivatives of functions like ax, sin x, and more.
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Presentation Transcript
First lets talk about what the integral means! Can you list some interpretations of the definite integral? b ( ) f x dx a
Heres a few facts: b ( ) f x dx 1. If f(x) > 0, then numerical value of the area between f(x) and the x-axis (area under the curve) a returns the a b = F(b) F(a) where F(x) is dx x f ) ( 2. any anti-derivative of f(x). (Fundamental Theorem of Calculus) a b 3. Basically gives the total cumulative dx x f ) ( change in f(x) over the interval [a,b]
What is a Riemann Sum? Hint: Here s a picture!
A Riemann sum is the area of n rectangles used to approximate the definite integral. = k 1 As n approaches infinity and = k 1 n = area of n rectangles ( ) f x x k k n b x dx ( ) ( ) f x f x k a So the definite integral sums infinitely many infinitely thin rectangles! (Calculus trivia: as n (number of rectangles) goes to the summation sign becomes the integral sign and x becomes dx)
The indefinite integral ( ) f x dx = ?
Wellhard to write; easy to say The indefinite integral equals the general antiderivative Where F (x) = f(x) ( ) f x dx = F(x) + C
+ 1 n nx 1 + xn dx = 1 + C
sin x dx Don t forget we are going backwards! So if the derivative was positive, the anti-derivative is negative. = - cos x + C cos x dx = sin x + C
1 dx = ln |x| +C x You need the absolute value in case x<0
1 dx where n > 1 xn Hint:
1/xn = x-n sooooooo . the answer is: + 1 n + C 1 + x 1 n You didn t say ln(xn) did ya??
Examples: dx 6 ( + ) 3 2 5 x x dx 3 x 4 dx ) 1 3 4 ( dx x 2+ 2 5 x dx dx +4 x 1 dx x
