What is the formula for the area between two curves y = f(x) and y = g(x) on [a,b] when f is above g?

What is the formula for the area between two curves y = f(x) and y = g(x) on [a,b] when f is above g?

• A. ∫_a^b (f(x) + g(x)) dx
• B. ∫_a^b (g(x) − f(x)) dx
• C. ∫_a^b (f(x) − g(x)) dx
• D. ∫_a^b (f(x) − g(x))^2 dx

Answer: (C)

Area between two curves is found by integrating the vertical gap between them across the interval. If f is on top, that gap is f(x) − g(x). So the area is ∫_a^b [f(x) − g(x)] dx. Since f(x) ≥ g(x) on [a,b], the integrand is nonnegative, giving the actual area value. The other forms don’t measure the gap correctly: adding the heights mixes in the total height rather than the distance between curves; subtracting in the opposite order would yield a negative value (unless you take an absolute value); and squaring the difference would give a quantity related to squared distance, not area. If you didn’t know which function is on top, you’d use ∫_a^b |f(x) − g(x)| dx.

Area between two curves is found by integrating the vertical gap between them across the interval. If f is on top, that gap is f(x) − g(x). So the area is ∫_a^b [f(x) − g(x)] dx. Since f(x) ≥ g(x) on [a,b], the integrand is nonnegative, giving the actual area value. The other forms don’t measure the gap correctly: adding the heights mixes in the total height rather than the distance between curves; subtracting in the opposite order would yield a negative value (unless you take an absolute value); and squaring the difference would give a quantity related to squared distance, not area. If you didn’t know which function is on top, you’d use ∫_a^b |f(x) − g(x)| dx.