How to solve a shaded region in a weird shape?

[Chaim]

Sorry, the left side is 3, and the right side is a 5
The base is 6, and the base of the shaded is 'x'.

What I was confuse with was how to start off?
The problem was: Express the area of the shaded region as a function of 'x'.
By the way, at the right bottom cornor is a right angle.

I know that x has to be greater than 6 and less than 6.
Though at first I thought I could've done something like 'x * 3'
Since it makes up most of the area there (but only makes up the rectangle)
So the slanted triangle wasn't included too.
Then I knew that the highest height was '5' so I did the area of a triangle
First I subtracted 3 (from the height on the left) from 5 (from the height on the right), to get the height of that triangle.
So I did the triangle formula: (1/2) base * height
(0.5)(x*2) = (0.5)(2x) = x

So I thought it was x * 3 to get the rectangle + x (from the area of the triangle)
So I got (x*3) + x, but this answer was wrong (I got the answer book)
So can anyone help me out here?

 

[daon2]

Take the lower part to be the x-axis, the left wall to be part of the y-axis. You have a shape bounded by the x axis and a line above it. The line has equation y=(1/3)x+3 (why?). The trapezoid is enclosed by the "bases": (1) the y axis with width 3, and (2) x, having width y= (1/3)x+3. The "height" of the trapezoid is x.

The area of a trapezoid is (1/2)(base1+base2)*height.

 

[galactus]

I used similar triangles by letting the height of the little shaded triangle be y. .

Per the diagram, \(\displaystyle \frac{2}{6}=\frac{y}{x}\)

\(\displaystyle y=\frac{x}{3}\)

The area of the little shaded triangle is \(\displaystyle \frac{xy}{2}\)

So, we get \(\displaystyle \frac{x(\frac{x}{3})}{2}=\frac{x^{2}}{6}\)

The area of the shaded rectangle is just 3x.

So, the shaded region has area \(\displaystyle 3x+\frac{x^{2}}{6}\)

Which you can rewrite how you wish.

But, if you wanted to use calculus, you could use the line Daon is mentioning and integrate from 0 to a.

\(\displaystyle \int_{0}^{a}(\frac{x}{3}+3)dx\)

Just replace the 'a' with an x when you're finished. Same result.