Final answer:
The area between two curves is found by integrating the difference of their functions over the bounds of intersection. This applies to one-dimensional cases like the work done by a force, and can also extend to other practical computations in physics and engineering such as surface and line integrals.
Step-by-step explanation:
The area of a region bounded by curves can be found by calculating the definite integral of the difference between the two curves within the bounds of intersection. In this context, if f(x) is the function of the upper curve and g(x) is the function of the lower curve, the integral representing the area between them from x1 to x2 is ∫ f(x) - g(x) dx from x1 to x2.
For the one-dimensional case, such as the work done by a force, the area under the force versus displacement curve can be calculated similarly by the integral. If the force is a linear function of displacement, such as f(x) = -kx, the area (in the context of work done) can be represented as a combination of the areas under the curve from a point xA to point xB, considering both positive and negative values of xA as in Equation 7.5.
Practical computation of surface integrals in physics or engineering applications involves taking the antiderivatives of the functions defining the surface and using the limits of the surface as bounds of the integral. This method also reflects how line integrals can be assessed, for example, when dealing with closed paths as described in Equation 8.8 and Equation 8.9. However, when evaluating a line integral, sometimes it's advantageous to express all variables in terms of a single parameter, such as either x or y, depending on the path and the functions involved.
The complete question is: Which of the following integrals represents the area of the region bounded by the curves? is: