Calculating the area under a curve is a standard problem in numerical methods. What you will develop is an app that calculates and displays the area under a range of curves. Figure 1: Area under the curve of y = x2 over the range 1 to 2, with 1 trapezoid. The area can be calculated by drawing one or more polygons (trapezoids) that approximate the curve. We start by drawing a trapezoid that encompasses our curve between the given limits and the x-axis, so for equation 1 of table 1, it looks like figure 1. We then calculate the area of the trapezoid. Notice that the trapezoid over-estimates the true area. With enough (smaller) trapezoids we can get a very good approximation to the area under the curve (see figure 2). The sum of the area of the smaller trapezoids is the area under the curve. The area of a trapezoid is given by: A = ½ (h1 + h2) d Plot graphs of the equations in table 1 for varying (input) total numbers of trapezoids. Compute and display the area between each curve and the x-axis. The number of trapezoids should be a positive integer. Extend your app by allowing the input of varying min and max x limits beyond those in Table 1, with max >= min Table 1: Sample curves and associated areas. Equation xMin xMax Actual Area y = x2 1 2 2 1/3 y = x2 – 4 -2 2 10 2/3 y = -x3 + 6x2 – x +17 2 4 80.0000 y = 2x3+2x2-5x+3 -2.5 7 65.2502 Figure 2: Area under the curve of y = x2 over the range 1 to 2, with 2 trapezoids. The more trapezoids, the better the approximation approaches the actual area. It is useful to know the error in the calculation. The relative error can be computed by: RE = | calculated area – actual area | actual area %RE = | calculated area – actual area | . 100% actual area Extend your app further by plotting %RE in area values (and the theoretical %RE line) for a range of trapezoids (1-100,000 in powers of 10 on the x-axis, 0.1-100 in powers of 10 on the y-axis) for each curve in Table 1. Question Week#9: App should allow input of the number of trapezoids and display correct plots of the curves for the data supplied in table 1 of the Problem Statement. Some unit tests implemented. Answer? I need answer from Solar2D and Lua( Special request don't do in any languages. Need in Solar2D and Lua and I need full code for this. (plz use Table 1) Please give proper explanation and typed answer only.
Calculating the area under a curve is a standard problem in numerical methods. What you will develop is an app that calculates and displays the area under a range of curves.
Figure 1: Area under the curve of y = x2 over the range 1 to 2, with 1 trapezoid.
The area can be calculated by drawing one or more polygons (trapezoids) that approximate the curve. We start by drawing a trapezoid that encompasses our curve between the given limits and the x-axis, so for equation 1 of table 1, it looks like figure 1. We then calculate the area of the trapezoid. Notice that the trapezoid over-estimates the true area. With enough (smaller) trapezoids we can get a very good approximation to the area under the curve (see figure 2). The sum of the area of the smaller trapezoids is the area under the curve.
The area of a trapezoid is given by: A = ½ (h1 + h2) d
Plot graphs of the equations in table 1 for varying (input) total numbers of trapezoids. Compute and display the area between each curve and the x-axis. The number of trapezoids should be a positive integer.
Extend your app by allowing the input of varying min and max x limits beyond those in Table 1, with max >= min
Table 1: Sample curves and associated areas.
|
Equation |
xMin |
xMax |
Actual Area |
|
y = x2 |
1 |
2 |
2 1/3 |
|
y = x2 – 4 |
-2 |
2 |
10 2/3 |
|
y = -x3 + 6x2 – x +17 |
2 |
4 |
80.0000 |
|
y = 2x3+2x2-5x+3 |
-2.5 |
7 |
65.2502 |
Figure 2: Area under the curve of y = x2 over the range 1 to 2, with 2 trapezoids.
The more trapezoids, the better the approximation approaches the actual area. It is useful to know the error in the calculation.
The relative error can be computed by:
RE = | calculated area – actual area |
actual area
%RE = | calculated area – actual area | . 100%
actual area
Extend your app further by plotting %RE in area values (and the theoretical %RE line) for a range of trapezoids (1-100,000 in powers of 10 on the x-axis, 0.1-100 in powers of 10 on the y-axis) for each curve in Table 1.
Question
Week#9: App should allow input of the number of trapezoids and display correct plots of the curves for the data supplied in table 1 of the Problem Statement. Some unit tests implemented.
Answer?
I need answer from Solar2D and Lua( Special request don't do in any languages. Need in Solar2D and Lua and I need full code for this. (plz use Table 1)
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