How to use Excel to find the approximation of definite integral?
Example 1: Use numerical methods to find the approximate value of
1 1 1 x 2 dx
Solution: The value of the definite integral equals the area between the graph of
1 x 2 and the x -axis ( x ranging from 1 to 1 ), which can be approximated by
calculating the sum of the rectangle’s area (as shown in the graph below).
Graph 1
Demonstraion: How to find the value of the sum using Excel? Exercise: What is the sum if we change x to 0.1?
Fact: By the Fundamental Theorem of Calculus, we have
x3 1 x dx F (1) F (1) , here F ( x) ( x c) 3
1 1 2
thus 1 1 1 x 2 dx (1
13 (1) 3 4 c) (1 c) . 3 3 3
You can also choose rectangles in this way:
Graph 2
Compare this graph with the first graph, the difference is: for each small interval, whether we choose the value at left end or right end as the height of the rectangle.
When you are finding the approximation of definite integral, you may take either the approach illustrated in graph 1 or graph 2.
Example 2: Use numerical methods to find the approximate value of
1 x 3 dx 0
Graph 3
In the graph above, x 0.1, and you may try different x . For the simplicity of calculation, just choose x such that the length of integration interval divided by x is an integer.
What is your result? If you choose x to be smaller and smaller, what value will you get? Numerically speaking, what do you think might the value of the indefinite integral?
Fact: By the Fundamental Theorem of Calculus, we have
x4 x dx F (1) F (0) , here F ( x) ( c) 4
1 0 3
14 04 1 thus 1 x 3 dx ( c) ( c) . 0 4 4 4