One of the motivations for the concept of integration is calculating areas. We know how to calculate areas of discrete shapes like rectangles, parallelograms, triangles, and from these we can figure out others. A goal is to calculate areas of continuously changing shapes like circles. If we are willing to accept approximations we can use discrete shapes to approximate the continuously changing shapes. Limits then provide a means to transition from appoximations to exact calculations.
For all of the images in Figure 4.1.1, compare the area between the parabola and the \(x\)-axis and the total area of the gray rectangles in that image.
As the number of rectangles increased in Figure 4.1.1 how did the approximation change? You can use Activity 29 to experiment. There are two major parts to this answer.
As the number of rectangles increased in Figure 4.1.2 how did the approximation change? You can use Activity 29 to experiment. There are two major parts to this answer.
At each of the \(x\)-coordinates listed draw a vertical line. Each line should go from \(y=0\) to \(y=1\text{.}\)\(x=-1,-\frac{3}{4},-\frac{1}{2},-\frac{1}{4},0,\frac{1}{4},\frac{1}{2},\frac{3}{4},1.\)
Notice that the vertical lines combined with the \(x\)-axis form three sides of a rectangle. For each of the \(x\) coordinates except \(x=2\) draw a top side on the rectangle starting at the height of the left segment. Notice that these are not all above and not all below the curve.
