Question

Consider the area between the graphs y=f(x) and y=g(x). This area can be computed in two different ways using integrals. First of all, it can be computed as a sum of two integrals.

a) Integrate ∫ᵇₐ[f(x)-g(x)]dx
b) Integrate ∫ᵇₐ[g(x)-f(x)]dx
c) Integrate ∫ᵇₐ[f(x)-g(x)]dx
d) Integrate ∫ᵇₐ[f(x)+g(x)]dx

Answer 1

To find the area between curves y=f(x) and y=g(x), integrate the difference between the functions from a to b. Use f(x)-g(x) when f(x) is above g(x), and g(x)-f(x) when g(x) is above f(x).

Step-by-step explanation:

To find the area between two curves, y=f(x) and y=g(x), using integrals, you typically take the integral of the difference between the two functions over the interval [a, b]. The area can be found through two approaches, depending on which function is on top:

1. Integrate from a to b the difference f(x) - g(x) when f(x) is above g(x).
2. Alternatively, integrate g(x) - f(x) when g(x) is above f(x).

Using any of these approaches will yield the absolute area between the two curves. You cannot use the integral of the sum f(x) + g(x) since that would not give the area between the curves, but rather combine the areas under both curves from the x-axis.