MATH1051 Lecture Notes - Lecture 3: 4Dx
5.2 – Definite Integral
Find area of between f(x)y= a≤x≤b
●Shortened: (x)dx
∫
b
a
f
Examples
●x 2 dx
∫
3
1
2 −
○You can either take the area after graphing it out, or take the antiderivative
(integral) directly
○ 2xx2−
■The antiderivative
○3) 2(3) 3( 2− =
■Plug in for b x
○1) 2(1) ( 2− = − 1
■Plug in for a x
○3) − ) 4( − ( 1 =
■Subtract from to find the area between the two pointsa b
The Integral using limit definition
●(x)dx (x)Δx
∫
b
a
f= lim
n→∞ ∑
n
k = 1
fk
●∑
n
k=1
1 = n
●∑
n
k=1
k=2
n(n+1)
●∑
n
k=1
k2=6
n(n+1)(2n+1)
●∑
n
k=1
k3=4
n(n+1)
22
These are a few of the definition used, but there is no need to memorize them.
Document Summary
Shortened: (x)dx between f(x) b a f. You can either take the area after graphing it out, or take the antiderivative (integral) directly x2 . The integral using limit definition to find the area between the two points (x)dx. = lim n n k = 1 f (x ) x k b f a n k=1 n k=1 n k=1 n k=1. These are a few of the definition used, but there is no need to memorize them. In this case, all three rectangle formulas would work, but right is the easiest so we"ll use that. Finding the above using what we"ve already solved for. We can substitute k with the equations listed above. The area under the curve is four, as it was in the previous problem lim n n n(n+1) n lim n . Properties of integrals a a b a b a f (x)dx. The integral of any function between intervals of same value is zero.
