Page 290 - Calculus Workbook For Dummies

274 Part V: The Part of Tens

If you already have, say, the midpoint approximation for ten rectangles and the trape-
zoid approximation for ten trapezoids, you can effortlessly compute the Simpson’s
Rule approximation for ten curvy-topped “trapezoids” with the following shortcut:
M n + M n + T n
S n2 = . This gives you an extraordinarily good approximation.
3

The Indefinite Integral

f x dx, is the family of all antiderivatives of f xh. That’s
The indefinite integral, # ^ h ^
why your answer has to end with “+ C.” For example, # 2 xdx is the family of all
2
2
2
2
parabolas of the form x + C like x - 1, x + 3, x + 10, and so on. All these are vertical
2
translations of y = x .
The Fundamental Theorem of Calculus, Take 1

Given an area function A f that sweeps out area under f t ^ h,
x
f ^
A x = # ^h f t dt,
h
s
the rate at which area is being swept out is equal to the height of the original function.
So, because the rate is the derivative, the derivative of the area function equals the
original function:
d A x = ^h f x . h
dx f ^

The Fundamental Theorem of Calculus, Take 2

Let F be any antiderivative of the function f ; then
b
f x dx F b -
# ^ h = ^ h F b ^ h.
a
The Definite Integral

b
f x dx, do is to add up an infinite number of
In essence, what all definite integrals, # ^ h
a
infinitesimally small pieces of something to get the total amount of the thing between
a and b. The expression after the integral symbol, f x dx (the integrand), is always a
^ h
mathematical expression of a representative piece of the stuff you’re adding up.

A Rectangle’s Height Equals

Top Minus Bottom

If you’re adding up rectangles with a definite integral to get the total area between two
curves, you need an expression for the height of a representative rectangle. This should
be a no-brainer: it’s just the rectangle’s top y-coordinate minus its bottom y-coordinate.