by interpreting it in terms of area
A definite integral can be interpreted as a net area, that is, as difference of the areas;
where A1 is the region above x- axis and below the graph of f, and A2 is the region below x- axis and above the graph of f.
i. Area of rectangle : A=l·b
ii. Area of quarter circle = A=14·π·r2
The integral ∫-301+9-x2dx can be interpreted as the area under the graph of
fx= 1+9-x2 between x= -3 and x=0
The graph of 1+9-x2 in the given domain is shown below.
From the graph,A1 represents the area of the rectangle and A2 represents one quarter of the area of circle with radius r=3.
Therefore, the area of a rectangle is
And thearea of aquarter circle is
Therefore, the net area is the addition of the two areas, since both are above the x axis