Mathematics

Mathematics

The insrtuction is:

f(x) = 4x^2 -x^3 is the function.

Find the definite integral over [0.4]. That will be your “exact area”.
Numerical Approach:
Midpoint Method
Trapezoid Method
Compute percentage area for each estimate:

% error = absolute value((exact-estimate)/exact)*100
Note: ABS() is used in excel to find absolute value

i Xi f(Xi) Xmid f(Xmid) ΔX M T S i Xi
0 0 0
1 1 0.5 1 1
2 2 1.5 1 2
3 3 2.5 1 3
4 4 3.5 1 4
Sum of areas = 5
6
7
i Xi f(Xi) Xmid f(Xmid) ΔX M T S 8
0 9
1 10
2 11
3 12
4 13
5 14
6 15
7 16
8

Numerical Integration

In order to compute an integral in closed form—that is, by using the Fundamental Theorem—you need to know two things:
• an explicit formula for the given function
• anantiderivative for the function.
For some applications, a function may consist of only a list of data rather than a formula. Even when a formula is known, sometimes no antiderivative exists. In such cases, the only way to compute the definite integral is by approximation. Variations on the methods used here are used by calculators and computers to evaluate definite integrals.
_____________________________________________________________________________
The Midpoint Rule
Suppose you want to integrate a function f(x) over the interval [a, b].
One of the simplest approximation methods can be visualized by
• subdividing the interval into n subintervals, each of width
• drawing thin rectangles of width Δx whose heights are determined by the height of the function’s graph at the midpoints of each of the subintervals. The midpoints can be denoted by .

• finding the sum of the areas of all the rectangles:

Exercise 1
Use the Midpoint Rule to approximate the area between the x-axis and the graph of from x = 0 to x = 4.

1. Open the file “Numerical Integration template” in Excel. The following information is already entered on Sheet 1:

This first Midpoint Rule approximation will use 4 rectangles, identified by number in the column headed i. The Xi column lists the x-coordinate at the center of each rectangle. The ΔX column lists the width of each rectangle.
2. Enter a formula in cell D3 that will calculate the height of the first rectangle using the x-coordinate in cell C3. (Remember to begin the formula with an = sign.) Copy this formula down through cell D6.
3. Enter a formula in cell F3 that will calculate the area of the first rectangle using the dimensions in cellsD3 and E3. Copy this formula down through cell F6.
4. Highlight cells F3 through F6. Then click on the “AutoSum” icon to see the sum of the rectangle areas in cell F7.
5. Use the Fundamental Theorem of calculus to find the exact value of the area under the curve.
6. Find the error and the percent error in the approximation.
7. On the same sheet, repeat the procedure, but this time use 8 rectangles.
8. Repeat once more, using 16 rectangles.
The Trapezoid Rule
As its name implies, this method uses a trapezoid instead of a rectangle as the basic unit.

Notice that x1 now identifies the right-hand edge of the first subinterval. This is because both of the vertical boundaries of each trapezoid are needed in order to find its area.
Exercise 2
1. The following information is already entered on Sheet 2:

Notice the extra row, which allows for the left-hand edge of the first trapezoid to be included.
1. As in Exercise 1, enter formulas in the D column that will compute the function values using the x-values in column C.
2. In cell F4, enter a formula that will compute the area of the first trapezoid. Copy the formula down the column. Then find the sum of the trapezoid areas.
3. Find the error and the percent error in the approximation.
4. On the same sheet, repeat the procedure, but this time use 8 trapezoids.
5. Repeat once more, using 16 trapezoids.
Simpson’s Rule
Both the Midpoint Rule and the Trapezoid Rule can give fairly good approximations to areas. Either one may be more accurate, depending on the area being analyzed.
Simpson’s Rule combines the advantages of both methods by using a weighted average of the two. If we let M represent the area of a rectangle found using the Midpoint Rule, and T the area of a trapezoid (with the same width) found using the Trapezoidal Rule, then the unit Simpson’s Rule area is found from the following formula:

Exercise 3
On Sheet 3, use Simpson’s Rule to approximate the same area you investigated in Exercises 1 and 2. The following information is already entered:

1. Complete the approximation for 4 subintervals. Then, as before, proceed to 8 subintervals and then to 16 subintervals. Compute the error and percent error each time.
2. Summarize your results, comparing the accuracy of the three approximation methods. You can enter your summary directly on Sheet 3.

Exercise 4
The figure below shows a tract of land with measurements in feet. A surveyor has measured its width w at 50-ft intervals (the values of x shown in the figure), with the following results:

Estimate the acreage of this tract using (a) the Trapezoidal Rule and (b) Simpson’s Rule. Use Sheet 4 of your Excel workbook. (Note: An acre is 43,560 ft2.)