Integration: Find the area between 2 lines.?

Question

Here is the question copied exactly:

The diagram shows a sketch of the curve y = sqrt (4 - x) and the line y = 2 - 1/3x. The coordinates of the points A and B where the curve and line intersect are (0,2) and (3,1) respectively. Calculate the area of the region between the line and the curve, giving your answer as an exact fraction.

The diagram shows the lines joining together at the y-axis, with the straight line having a slightly larger x value on the x-axis. The curve has a greater y-value than the line up until they meet at point B(3,1).

The answer given is 1/6, but I can't seem to be able to get this answer... the answer given could be wrong, but I think it's more likely that I'm the one who is wrong.

Any help is greatly appreciated.

Answer 1

Area = ∫(√(4-x) - (2 - x/3))dx from 0 to 3.

...... = [(-2/3)(4 - x)^(3/2) - 2x + x²/6] from 0 to 3

...... = [(-2/3)*1 - 2*3 + 9/6] - [(-2/3)4^(3/2)]

...... = [ -2/3 - 6 + 3/2] - [(-2/3)*8]

...... = -31/6 + 16/3

...... = 1/6