Midpoint Rule with EXCEL
Find 
(the integral of 1/(x2+5) from 1 to 4) by the
midpoint rule with n = 20, using an EXCEL worksheet.
The problem can be solved with the following worksheet (the formulas are shown below):
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dx=(b-a)/n |
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0.15 |
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dx/2 |
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The initial x is a+dx/2, where a is the lower limit of integration. The second and subsequent x's
are formed by adding dx to the previous x To extend formulas downward, move the cursor to
the lower right corner of the cell to be extended until it changes to a
solid +, click and hold with the mouse, and drag downward |
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0.075 |
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f(x) |
integral |
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n |
x |
1/(x^2+5) |
dx * sum of
f(x) |
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1 |
1.075 |
0.162453 |
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2 |
1.225 |
0.153831 |
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3 |
1.375 |
0.145125 |
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4 |
1.525 |
0.136507 |
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5 |
1.675 |
0.128113 |
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6 |
1.825 |
0.120039 |
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7 |
1.975 |
0.112352 |
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8 |
2.125 |
0.10509 |
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9 |
2.275 |
0.098274 |
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10 |
2.425 |
0.091906 |
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11 |
2.575 |
0.08598 |
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12 |
2.725 |
0.080479 |
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13 |
2.875 |
0.075383 |
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14 |
3.025 |
0.070668 |
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15 |
3.175 |
0.06631 |
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16 |
3.325 |
0.062283 |
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17 |
3.475 |
0.058563 |
Compare this midpoint rule approximation with
the actual value of .286450284649, obtained from a graphing calculator. |
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18 |
3.625 |
0.055125 |
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19 |
3.775 |
0.051946 |
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20 |
3.925 |
0.049006 |
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sum: |
1.909434 |
0.286415161 |
The formulas I used to generate the worksheet are as follows:
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dx=(b-a)/n |
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=(4-1)/20 |
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dx/2 |
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=A2/2 |
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f(x) |
integral |
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n |
x |
1/(x^2+5) |
dx * sum of
f(x) |
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1 |
=1+A4 |
=1/(B7^2+5) |
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=1+A7 |
=B7+0.15 |
=1/(B8^2+5) |
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=1+A8 |
=B8+0.15 |
=1/(B9^2+5) |
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=1+A9 |
=B9+0.15 |
=1/(B10^2+5) |
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=1+A10 |
=B10+0.15 |
=1/(B11^2+5) |
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=1+A11 |
=B11+0.15 |
=1/(B12^2+5) |
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=1+A12 |
=B12+0.15 |
=1/(B13^2+5) |
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=1+A13 |
=B13+0.15 |
=1/(B14^2+5) |
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=1+A14 |
=B14+0.15 |
=1/(B15^2+5) |
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=1+A15 |
=B15+0.15 |
=1/(B16^2+5) |
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=1+A16 |
=B16+0.15 |
=1/(B17^2+5) |
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=1+A17 |
=B17+0.15 |
=1/(B18^2+5) |
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=1+A18 |
=B18+0.15 |
=1/(B19^2+5) |
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=1+A19 |
=B19+0.15 |
=1/(B20^2+5) |
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=1+A20 |
=B20+0.15 |
=1/(B21^2+5) |
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=1+A21 |
=B21+0.15 |
=1/(B22^2+5) |
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=1+A22 |
=B22+0.15 |
=1/(B23^2+5) |
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=1+A23 |
=B23+0.15 |