Question What is the net signed area between f(x) = 2x + 3 and the x-axis over the interval (-4,2]? Do not include any units in
your answer
Provide your answer below:

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!

8

step-by-step explanation: