Math 1A Study Group
Faculty: Ole Hald
Study Group Leader: Larry X. Wang, larry@csrjjsmp.com
Location: MW 11-12 201A Chavez
Community through Academics and Leadership
Worksheet #17: Introduction to Integrals
1. Express the following in Reimann sum notation:
a) 1+3+5+7+…+53
b) 1+2+4+8+…+64
1 1
1 1
c)  1    ...  
2 3
9 10
2. Prove by induction:
n
n(n  1)
a)  i 
2
i 1
n
b)
 2i  1  1  3    (2n  1)  n
2
i 1
n
c)
i
n
 ( i ) 2
3
i 1
i 1
d) 8 n  1 is divisible by 7
2
3. Let A   5 x  3dx . First, let n  5 , and estimate A with each method.
1
a)
b)
c)
d)
e)
f)
g)
h)
4.
Use the left endpoints. Is the approximation an overestimation or underestimation of A?
Use the right endpoints. Is the approximation an overestimation or underestimation of A?
Use the midpoints.
Use the points halfway between the left endpoints and the midpoints.
Find the difference between the area estimated by the right endpoints and the area estimated
by the left endpoints.
Now let n  10 . Find the difference between the area estimated by the right endpoints and
the area estimated by the left endpoints.
Find the difference between the area estimated by the ¼-points and the area estimated by the
¾-points.
Find the difference between the area estimated by the right endpoints and the area estimated
by the left endpoints as n approaches infinity.
Find an integral that would equal to the given limit.
3
4 4

a) lim  12  i 
n 
n n
i 1 
n
 

b) lim  cos  i 
n 
3n  3n
6
i 1
5. Find the integral by the limit definition.
n
4
a)
 (x
2
 3 x )dx
1
3
b)
 (x
3
4
 x 2  1)dx