Okay, so the graph they've drawn is for the function - don't even think about t or the shading yet, build things up little by little.

Note that it's a hybrid, or piece-wise, function. So, to draw the graph, you start by drawing the first part. Draw the graph of y=3x. Ignore everything else, just draw that graph. Now, draw the graph of y=3 - that's the second part of our function. Now that you've done this, look at the domain of each part. The first part, y=3x, is only defined for when 0<=x<=1. So, whenever x ISN'T between 0 and 1, erase the line y=3x. That's what gives you the first part of the graph, where is a rising line. For the next part, y=3, is only defined when x>1 - so erase every part of the graph where x ISN'T greater than 1. That's what gives you the flat line in their graph.

Now that you've drawn the piece-wise function, it's time to get to the actual question - what's the area under the curve up to the line x=t? Well, remember those questions in year 10 measurement, where they'd give you those weird shapes you didn't have a formula for? And you had to calculate the area by cutting up the weird shape into parts?

For example, say you have an L-shaped block. To calculate the area, you want to cut the | from the _ (L=|_, can you see it?). Now, it's not the area of a shape you don't have the formula for - it's the area of two rectangles!

Well, we're going to do the same thing here - the total area is going to be the area of the triangle in the first part of the function (y=3x) and the area of the rectangle after that (y=3). So, what's the area of this?

Well, the area of the triangle is (1/2)*b*h. The area of the rectangle is l*w. The question is, what are all these values? You should be able to find them by looking at your graph.

Now, why did they draw two with two different shadings? Well - if the value of t is less than 1, then the line won't include the rectangle, so it'll just be the area of the triangle. Again, you should be able to figure out this area by looking at your graph. Remember, we don't know WHAT value t is - just that at this point, it's less than 1. I encourage you to try playing around with the graph now that you (hopefully) understand it to try and calculate the area for some unknown value of t