Demonstrate understanding of parts of definition of integral using an application
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.
Subsection4.1.1How Square Pegs Fill Round(ed) Holes
Activity4.1.1.Comparing Approximations.
The goal of this activity is to compare the values of area approximations to actual areas and to compare approximations to each other.
The function is \(f(x)=2x-x^2\text{.}\)
(a)
For each image in Figure 4.1.1, calculate the total area of the gray rectangles.
(b)
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.
(c)
For each image in Figure 4.1.2, calculate the total area of the gray rectangles.
(d)
For each image in Figure 4.1.2, compare the area between the parabola and the \(x\)-axis and the total area of the gray rectangles in that image.
(e)
As the number of rectangles increased in Figure 4.1.1 how did the approximation change? You can use Activity 4.1.1 to experiment. There are two major parts to this answer.
(f)
As the number of rectangles increased in Figure 4.1.2 how did the approximation change? You can use Activity 4.1.1 to experiment. There are two major parts to this answer.
(g)
How could the approximation of the area between the parabola and the \(x\)-axis be improved? What calculus concept does this require?
Figure4.1.1.Approximations of Area 1
Figure4.1.2.Approximations of Area 2
Instructions.
Select various values of \(n\text{.}\) Compare the results of the approximations — labeled lower sum and upper sum — to the actual area.
Figure4.1.3.Experimenting with Approximations of Area
Subsection4.1.2Constructing an Area Approximation
Activity4.1.2.Constructing an Area Approximation.
This activity provides directions for constructing an area approximation (called Riemann Sum)
(a)
Sketch the curve \(y=\sqrt{1-x^2}\) on its whole domain \((-1,1)\text{.}\)
(b)
At each of the \(x\)-coordinates listed draw a vertical line. Each line should go from \(y=0\) to \(y=1\text{.}\)\(x=-1,-\frac{3}{4},-\frac{1}{2},-\frac{1}{4},0,\frac{1}{4},\frac{1}{2},\frac{3}{4},1.\)
(c)
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.
(d)
Calculate the total area of the set of rectangles.
