Answer:
Step-by-step explanation:
2 units squared.
Explanation:
You can go about this in two ways: Algebra and Integrals.
Algebra
y = 2x-6 is linear, thus if we take the value over the x interval [2,4] we can use geometry to calculate the area.
graph{2x-6 [-2.27, 7.73, -2.18, 2.82]}
You can see two triangles in the graph (you could also find this algebraically). Thus, you can calculate the area.
=
2
⋅
(
1
2
⋅
b
⋅
h
)
=
2
⋅
(
1
2
⋅
1
⋅
2
)
=
2
square units
Integrals
An integral gives the area under the curve. Remember though that if a function goes below the X axis, the integral is negative Thus you have to do two separate integrals based on the X intercept.
finding the X intercept (that is where y = 0)
2
x
−
6
=
0
x
=
3
when
y
=
0
When
x
<
3
, then y is negative. Thus, we have to find the negative integral from 2 to 3 and the positive integral from 3 to 4
=
−
∫
3
2
2
x
−
6
d
x
+
∫
4
3
2
x
−
6
d
x
=
−
[
x
2
−
6
x
]
3
2
+
[
x
2
−
6
x
]
4
3
=
−
[
9
−
18
−
4
+
12
]
+
[
16
−
24
−
9
+
18
]
=
−
[
−
1
]
+
[
1
]
=
2
units squared. And Tada, that is the same as the other answer!
