C2 - Definite Integration

Hi, our class were taught integration last week.

I'm really confused about negative area, particularly this question. This may be my arithmetic, but I'm not sure why I got "negative" area for the curve of 4 - x^2 and positive area for (1/2)4 - x^2.

Attached is the question (16) and my working.

Can someone guide me through:

1. This question and any mistakes I made.

2. The concept of negative area, as all I know is that it is because it is under the x-axis, but without any proof as to why it is "negative" area.

Kind regards,

Kevin

Your limits are wrong; the limits should be the range of x-values not the range of y-values.

The sign of the area makes a difference sometimes when the integral is in some context but for a simple area problem such as this I would not overthink it too much; a negative area is one below the x-axis.