math pracs-integral as limit of sum

1. Math practical-4
Class 12

2. Experiment no. 4
Draw this figure here
• Aim: To evaluate the definite integral of the given curve within the
given limits as the limit of sum and verify it by actual integration.
• Software requirements: Geogebra.
• Other requirements: print out of the graph in the applet, external
usb memory
• First Fundamental theorem of integral calculus: let f be a
continuous function on[a,b] and let A(x) be the area function. Then
A’(x) = f(x) ∀ 𝑥 ∈ [𝑎, 𝑏]
• Second Fundamental theorem of integral calculus: let f be a
continuous and defined function in [a,b] and F be the antiderivative
of f. Then, 𝑎
𝑏
𝑓 𝑥 𝑑𝑥 = [𝐹 𝑥 ] 𝑎
𝑏
= 𝐹 𝑏 − 𝐹(𝑎)
• Formula used: 𝑎
𝑏
𝑓 𝑥 𝑑𝑥 = h lim
𝑛→∞
{𝑓 𝑎 + 𝑓 𝑎 + ℎ + 𝑓( 𝑎 +
Upper sum
Lower sum

3. Method followed
1. Open the applet “integral as limit of sum” and save it in the external memory with
your name as “Name_ integral as limit of sum” .
2. Work on this file now
3. Observe that the function 𝑓 𝑥 =
1
4
𝑥3
− 𝑥2
+ 𝑥 + 2 is a real valued function
defined in [1,4].
4. Observe the continuity of the function f(x) in [1,4]
5. Observe the differentiability of the function f(x) in (1,4).
6. Ensure that ‘a’ , ‘b’ and ‘c’ on the left hand side panel are are unseleted.
7. Select ‘a’ and record 8 observations by varying n from 1 to 100, at a difference of 10
each. For eg. Observation 1 is for n=5, observation 2 for n=15 observation 3 for n= 25
and so on.
8. Now in the left panel, ensure that the point on the curve ‘P’ is selected.
9. Choose any convenient value of n, say 10 (or more than 10). now record the height
of each rectangle by moving the point P on the curve from x=1 to 4.
10. Record the width of each rectangle.
11. For this value of n, calculate the area of each rectangle and add them.
12. Select ‘c’ on the L.H.S. panel. Observe the value of ‘c’; the actual area under the
curve and find the difference between the values of ‘a’ and ‘c’
13. Now unselect ‘a’ and ’c’ and select ‘b’ on the L.H.S. panel.
14. Repeat the 7th to 13th step to record 10 observations for different values of n.
(choose the same set of values for n).
S. No. Value of n Value of ‘a’ Absolute Difference
between value of a
and c
1
2
3
4
5
6
7
8
Observations
Table for ‘a’
Height of
rectangle
Width of
rectangle
Area of
rectangle
1
2
3
4
5
6
7
8
9
10
Sum of the areas
Observation table: Areas of rectangles for n=___

4. Demonstration
1. 𝑎
𝑏
𝑓 𝑥 𝑑𝑥 = [𝐹 𝑥 ] 𝑎
𝑏
= 𝐹 𝑏 − 𝐹(𝑎), is called ___________ and here,F(x)
is called ___________
2. To apply the above, the condition on f(x) is _______________
3. “a” denotes ______, ‘b’ denotes______ and ‘c’ denotes ______
4. When n is 1, the difference between a and b is _____, difference between a
and c is _____ and difference b and c is _____
5. When n is 1, the difference between a and c is much more than the difference
between b and c because ________
6. As n increases, the difference between a and b _________. The difference
between a and c ______. The difference between b and c ______
7. The lower sum b is always ______the actual sum c and the upper sum a is
always ________the actual sum c.
8. When n = 100, the absolute difference between a and b is __ between a and c
is ___ and between b and c is _____
9. If n is made larger, do you think, the value of a,b and c will become equal?
______ why?________
10. Mathematical verification of the value obtained above by integrating the given
function between the given limits:
Conclusion
For n =100, a = ______, b=____ c=_______. The value obtained by mathematically
solving the given integral is ___________
So, as 𝑛 → ∞, 𝑎 → 𝑐 𝑎𝑛𝑑 𝑏 → 𝑐. And the value of c is equal to the mathematically
obtained value
Observation
Table for ‘b’
S. No. Value of n Value of ‘b’ Absolute Difference
between value of b
and c
1
2
3
4
5
6
7
8
Height of
rectangle
Width of
rectangle
Area of
rectangle
1
2
3
4
5
6
7
8
9
10
Sum of the areas
Observation table: Areas of rectangles for n=___

5. Precautions
1. Copy the file in an external memory and work on that file,
don’t save your work on the desktop file.
2. Don’t tamper with the in build functions of the applet ; rather
use a fresh file to make a new applet if needed.
3. Save the file with a different name in the external memory,
don’t overwrite on the desktop file.
4. While taking a print out of the graph applet, change the
author’s name to your own name.
Note :
1. you may download geogebra classic 5 from
https://www.geogebra.org/download
2. You may get a copy of this applet from
https://drive.google.com/file/d/1VR7fWRXC8BJcAHG
DhlKTveIy3uzxOEyj/view?usp=sharing
3. For all the practicals, visit
https://sites.google.com/view/mathpractical
• Paste a print out here for upper and lower sum integral each
for one of the values of n taken for observation.