Final answer:
To find the area between curves y=f(x), y=g(x), and vertical lines x=a and x=b, integrate each function from a to b and take the difference of the results as \(\int_a^b (f(x)-g(x))dx\), taking into account any intersections within the bounds.
Step-by-step explanation:
To find the area of the region enclosed between the curves y=f(x), y=g(x), and the vertical lines x=a and x=b, follow these steps:
First, sketch the functions f(x) and g(x) to determine their points of intersection and to visualize the enclosed area.
Calculate the integral of f(x) from x=a to x=b, which gives the total area under the curve f(x) between these bounds.
Do the same for g(x), finding the integral of g(x) from x=a to x=b.
The enclosed area is the difference between the two integrals, assuming f(x) is above g(x) on the interval [a,b]. Write this as \(\int_a^b (f(x)-g(x))dx\).
If f(x) and g(x) intersect within the limits of integration, the calculation may involve multiple integrals split at the points of intersection.
For example, if h(x) represents the height of a rectangle that could be used as an approximation for the area under a curve, the area can similarly be gauged by integrating the function over the desired interval.
In physics, a similar method is used to calculate work done, where the area under the force vs. displacement graph gives the work, reminiscent of finding the enclosed area in a mathematical function.