MATH1051 Lecture 7: Lecture 07
6.1 – Area Between Curves
Find the Area of the Region
●What we want to do is take the integral, which would give us the area under the curve
●However, if we were to take the integral of or only, you wouldn’t quite get(x)g(x)f
the area highlighted in green that we want
○What to do?
○Take the area of (x) g(x)f−
○Why? Because the area under the curve of subtracted from the area under(x)g
the curve of would give us the desired area.(x)f
●What would this give us?
○ (x) g(x)dxA = ∫
b
a
f−
○ OP (x) BOT (x)dxA = ∫
b
a
T−
■In other words, the area between the curves is the area of the top curve
minus the bottom curve
Example #1
● Find the area between and 2x y= xy = 2
●Firstly, we want to define the boundaries that we will take the area under
○Find the intersection points of the two functions
○Set the functions equal to each other to do so
○x x2 = 2
○ x 2x0 = 2−
○ x(x 2)0 = −
○, 2x= 0x=
○So we will take the integral from [0, 2]
●x x dx
∫
2
0
2 − 2
○We know that we can compile the new integral like so because from the graph,
we can clearly see that is clearly on the top 2xy =
● x x dxA = ∫
2
0
2 − 2
○x(− x) = 23
13
○2 2 ) (0 0 )( 2−3
13− 2−3
13
○ = 3
4
Example #2
●Find the area between the curves 4, y 2 , x 1y= = x =
Document Summary
What we want to do is take the integral, which would give us the area under the curve. However, if we were to take the integral of only, you wouldn"t quite get (x)g or f (x) the area highlighted in green that we want. Take the area of (x) f g(x) Because the area under the curve of (x)g subtracted from the area under the curve of (x) What would this give us? f would give us the desired area. b. In other words, the area between the curves is the area of the top curve minus the bottom curve. Firstly, we want to define the boundaries that we will take the area under y = 2 y = and. Find the intersection points of the two functions. Set the functions equal to each other to do so x x. So we will take the integral from [0, 2]
