Sketch the graphs of the functions.

Question

f(x) = −x² + 3x and g(x) = x − 3

Find the area of the region completely enclosed by the graphs of the given functions f and g.

Answer 1

As f(x) is a parabola facing down and g(x) is a line, we shall subtract g(x) from f(x) and hope for zeroes.

f(x) - g(x) = -x² + 3x - (x - 3) = -x² + 2x + 3 = -1(x-3)(x+1). We have 0s at -1 and 3, so we will integrate on [-1,3]

Using I[a, b] for the integral from a to b and E[a,b] for the evaluation from a to b,

I[-1, 3] -x² + 2x + 3 dx = -x³/3 + x² + 3x E[-1, 3] = -27/3 + 9 + 9 - (1/3 + 1 - 3) = 10 2/3