MATH103 Openstax Calculus Volume 2 Solution Manual 2026

,OpenStax Calculus Volume 2 Student Answer and Solution Guide


Chapter 1
Integration
1.1. Approximating Areas

Section Exercises

1. State whether the given sums are equal or unequal.
10 10
a.  i and
i =1
k
k =1
10 15
b.  i and
i =1
 ( i − 5)
i =6
10 9
c.  i ( i − 1) and  ( j + 1) j
i =1 j =0


 i ( i − 1) and  ( k −k)
10 10
2
d.
i =1 k =1
Answer: a. They are equal; both represent the sum of the first 10 whole numbers. b. They are
equal; both represent the sum of the first 10 whole numbers. c. They are equal by substituting
j = i − 1 . d. They are equal; the first sum factors the terms of the second.


In the following exercises, use the rules for sums of powers of integers to compute the sums.

10

åi 2

3. i=5

Answer: 385 − 30 = 355

100 100
Suppose that  ai = 15 and
i =1
 b = −12 . In the following exercises, compute the sums.
i =1
i



å( a - b )
100

i i
5. i=1

Answer: 15 − ( −12 ) = 27


( )
100

å 5ai + 4bi
7. i=1

Answer: 5 (15 ) + 4 ( −12 ) = 27

,OpenStax Calculus Volume 2 Student Answer and Solution Guide


In the following exercises, use summation properties and formulas to rewrite and evaluate
the sums.


å( j )
50
2
-2j
9. j=1

50 50
( 50 )( 51)(101) − 2 ( 50 )( 51)
Answer:  j =1
j 2 − 2 j =
j =1 6 2
= 40,375



å éêë( 2k )
25
-100k ùú
2


11. k=1
û
25 25 4 ( 25 )( 26 )( 51)
Answer: 4 k 2 − 100 k = − 50 ( 25)( 26 ) = −10,400
k =1 k =1 9


Let Ln denote the left-endpoint sum using n subintervals and let Rn denote the
corresponding right-endpoint sum. In the following exercises, compute the indicated left
and right sums for the given functions on the indicated interval.

g ( x ) = cos (p x )  0, 1
13. R4 for on
Answer: R4 = 0.25

1
15. R6 for f ( x ) = on  2, 5
x ( x − 1)
Answer: R6 = 0.372


on  −2, 2 
1
17. L4 for
x +1 2


Answer: L4 = 2.20

19. L8 for x2 − 2 x + 1 on  0, 2
Answer: L8 = 0.6875

21. Compute the left and right Riemann sums—L6 and R6, respectively—for
f ( x ) = ( 3 − 3 − x ) on  0, 6 . Compute their average value and compare it with the area under
the graph of f.
Answer: L6 = 9.000 = R6 . The graph of f is a triangle with area 9.

, OpenStax Calculus Volume 2 Student Answer and Solution Guide


23. Compute the left and right Riemann sums—L6 and R6, respectively—for
f ( x ) = 9 − ( x − 3) on  0, 6 and compare their values.
2



Answer: L6 = 13.12899 = R6 . They are equal.


Express the following endpoint sums in sigma notation but do not evaluate them.

f ( x) = 4 - x2  −2, 2
25. L10 for on
4 10  (i − 1) 
Answer: L10 = 
10 i =1
4 −  −2 + 4
 10 



27. R100 for ln x on 1, e 
e − 1 100  i 
Answer: R100 =  ln 1 + ( e − 1)
100 i =1 

100 


In the following exercises, graph the function then use a calculator or a computer program
to evaluate the following left and right endpoint sums. Is the area under the curve between
the left and right endpoint sums?

29. [T] L100 and R100 for y = x 2 on the interval  0,1
Answer:




R100 = 0.33835 , L100 = 0.32835 . The plot shows that the left Riemann sum is an underestimate
because the function is increasing. Similarly, the right Riemann sum is an overestimate. The area
lies between the left and right Riemann sums. Ten rectangles are shown for visual clarity. This
behavior persists for more rectangles.