The easiest way to answer this is to use an example.
You will need a function that defines the curve and you will need to know the limits of the definite integration. Let’s call the limits lo and hi.
If you have a function f(x) then you will need a 4-column table. The function may be given as y= then an expression containing x, that is, y=f(x).
Label the 4 columns “x”, “∆x” (or “dx”), “f(x)” (or “y”), and “f(x)∆x” (or “f(x)dx”, etc.).
You need to decide how many subdivisions n of the interval [lo,hi] you need. The greater the value of n the more accurate you can expect the result to be. Let’s say the interval is [1,3] divided into 20 subdivisions, so ∆x=(3-1)/20=0.1.
In general, ∆x (hi-lo)/n.
In the first cell under x type in the value corresponding to lo. In the first cell under ∆x, enter ∆x. These are the only two absolute values you need. All the other cells will contain formulas.
Under f(x) you will need to write the formula using the cell reference under the x column. For example, if f(x)=x³+2x²+3x+4, then the Excel cell will be:
=A2^3+2*A2^2+3*A2+4
where A2 is the reference of the first cell under column x on your spreadsheet. (The equals must be inserted at the beginning to tell Excel that you are submitting a formula.) Let’s say that you entered this formula into cell C2 on your spreadsheet. (B2 contains the value for ∆x.)
Under f(x)∆x you enter the formula:
=B2*C2
In A3 enter =A2+B2 and in B3 enter =B2 (don’t forget the equals).
Drag your cursor over cells A3 and B3 to select them, then choose Copy. Excel will highlight them with a dotted boundary. Now select by dragging your cursor over cells in the A and B columns scrolling down as you drag until you’ve reached down roughly as far as you expect. Paste the copied cells into this region. You don’t need to be accurate at this stage about how far down to go.
Next, starting at C2 and D2, select these cells, copy them and paste into a region below more or less in line with those in columns A and B.
If you’ve got this right you will see calculations over all four columns.
You may need to adjust the column widths so that you can see the figures to a sufficient number of decimal places. You can do this at any time. Excel works to about 10 significant figures accuracy.
To extend the number of rows in your 4-column table you can copy and paste in blocks of 4-column widths until you observe the value of hi in the x column.
Any surplus rows can be cleared by selecting and using clear.
Finally, use the SUM command or facility from the menu bar to drag-select from D2 to the first empty cell in the D column following the column of figures. (The SUM command is, for example:
=SUM(D2:D202)
where D202 is the call reference of the last cell in the column containing figures.
This will be the result of the definite integration.
