Area Bounded by Two Functions at Interval [1, 9]

Question

Find the area of the region enclosed by f(x) = √x and g(x) = -(1/2)x - 3 and the vertical lines x = 1 to x = 9.

Answer 1

Draw a diagram. Between x = 1 and x + 9, f(x) > g(x).

So, Area = ∫_{(1 to 9)}[f(x) - g(x)]dx

= [(2/3)x^3/2 + (1/4)x² + 3x] _{(from 1 to 9)}

= [18 + 81/4 + 27] - [2/3 + 1/4 + 3] = 45 + 81/4 - 47/12 = 45 + 196/12 = 45 + 16 1/3 = 61 1/3 = 184/3