Use geometry(geometric formulas) to evaluate the integral from 0 to 6 f(x) dx, for the system of equations:

f(x)= 4-x for x LESS than or EQUAL to 4 and
equals 2x-8 for x > 4.

A. 24
B.8
C.16
D.12

I could really use some help please!
Thank you!

Did you make your sketch?

Mine has two right-angled triangles.
1. for f(x) = 4-x from 0 to 4 --- triangle with vertices (0,0), (4,0) and (0,4)
2. for f(x) = 2x - 8 for x>4 ---- triangle with vertices (4,0), (6,0), and (6,4)

Area = (1/2)(4)(4) + (1/2)(2)(4) = 12

If we want to use "overkill" and apply Calculus , we would get
∫ (4-x) dx from 0 to 4 + ∫ (2x-8) dx from 4 to 6
= 4x - x^/2| from 0 to 4 + x^2 - 8x | from 4 to 6
= (16 - 8 - (0-0) ) + (36 - 48 - (16 - 32) )
= 8 + 4
= 12

Yes, I know that the definite integral gives me the area under the curve for f(x). BUT...... my only problem is that I don't understand how to do this problem at all because in my module/chapter, I've never ran in to a problem like this using a system of equations. This is legit the first time they've given me this kind of problem.

go for it

Similar to Reiny:

Draw a Cartesian coordinate system x = 0 to 6

Draw line y = 4 - x , where x = 0 to 4

For x = 0 , y = 4 , for x = 4 y = 0

Draw line y = 2 x - 8 , where x = 4 to 6

For x = 4 , y = 0 , for x = 6, y = 4

The definite integral is area of the region bounded by the graph and x-axis.

In this case:

A1 = one half area of square ( 4 * 4 ) betwen x = 0 and x = 4

and

A2 = one half area of rectangle ( 2 * 4 ) betwen x = 4 and x = 6

A = A1 + A2

A = 4 ∙ 4 / 2 + 2 ∙ 4 / 2

A = 16 / 2 + 8 / 2

A = 8 + 4 = 12

Answer 12

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Remark:

You have A2 = one half area of rectangle ( 2 * 4 ) becouse:

one side of rectangle:

x = 6 - 4 = 2

other side of rectanlge:

y = 4 - 0 = 4
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@Reiny do you mind helping me out on one more question please?

Thank you.

Okay I posted it. Thank you sooo much man! I really do appreciate it!

This is not a "system" of equations.

Remember back in pre-cal, where you studied piece-wise functions?

To evaluate the given integral, we need to divide the range of integration (0 to 6) into two parts based on x = 4, where the function changes.

For x ≤ 4, the function is f(x) = 4 - x.
For x > 4, the function is f(x) = 2x - 8.

Let's evaluate the integral using geometric formulas:

1. For the part of the integral from 0 to 4:
We have f(x) = 4 - x.
The area under this curve is the area of a trapezoid with base 1 (horizontal side) and height (vertical side) 4.
The formula for the area of a trapezoid is A = (b1 + b2) * h / 2, where b1 and b2 are the parallel bases and h is the height.
Plugging in the values, we get A = (1 + 4) * 4 / 2 = 10.

2. For the part of the integral from 4 to 6:
We have f(x) = 2x - 8.
The area under this curve is the area of a triangle with base 2 (horizontal side) and height (vertical side) 2.
The formula for the area of a triangle is A = base * height / 2.
Plugging in the values, we get A = 2 * 2 / 2 = 2.

Now, we sum up the two areas: 10 + 2 = 12.

Therefore, the value of the integral from 0 to 6 f(x) dx is 12. So, the correct answer is D. 12.

You do know that a definite integral gives you the area under the curve for f(x), right?

Did you draw the graph? All you have to do is figure the area of two triangles.