Work done by a variable force

The basic work relationship W=Fx is a special case which applies only to constant force along a straight line. That relationship gives the area of the rectangle shown, where the force F is plotted as a function of distance. In the more general case of a force which changes with distance, the work may still be calculated as the area under the curve. For example, for the work done to stretch a spring, the area under the curve can be readily determined as the area of the triangle. The power of calculus can also be applied since the integral of the force over the distance range is equal to the area under the force curve: For any function of x, the work may be calculated as the area under the curve by performing the integral 基本的功关系式 W=Fx 是一个特殊情况，仅适用于沿直线作用的恒力。该关系式给出了图中所示的矩形面积，其中力 F 作为距离的函数进行绘制。在更一般的情况下，当力随距离变化时，功仍可以计算为曲线下的面积。例如，对于拉伸弹簧所做的功，曲线下的面积可以很容易地确定为三角形的面积。由于力在距离范围内的积分等于力曲线下的面积，因此微积分的威力也可以得到应用：对于任何 x 的函数，通过进行积分，可以计算出功为曲线下的面积。

Work: General Definition

The general definition of work done by a force must take into account the fact that the force may vary in both magnitude and direction, and that the path followed may also change in direction. All these things can be taken into account by defining work as an integral.