CEGEP CHAMPLAIN - ST. LAWRENCE
201-203-RE: Integral Calculus
Patrice Camiré
Problem Sheet #3
Riemann Sums, Area and the Definite Integral
1. Write the sum in expanded form.
(a)
5
X
√
3i − 1
6
X
(−1)i
(b)
i2 + 1
(c)
i=0
i=1
4
X
j
2
(d)
n
X
3k
k=0
j=1
2. Write the sum using sigma notation.
(a) 9 + 11 + 13 + 15 + 17 + 19 + 21
2 4 6 8 10
(b) − + − + −
3 5 7 9 11
(c) 1 − 4 + 9 − 16 + 25 − 36 + 49
sin(2) sin(3) sin(4)
(d) sin(1) + √ + √
+ √
3
4
2
3
4
3. Write down the value of the following sums where n is a positive integer.
(a)
n
X
1
(b)
i=1
n
X
i
(c)
i=1
n
X
i2
(d)
i=1
n
X
i3
i=1
4. Evaluate the following sums.
(a)
(b)
(c)
100
X
1
(d)
30
X
i2
(g)
15
X
(3i2 − 5)
(j)
14
X
i=1
i=1
i=1
i=1
50
X
17
X
12
X
14
X
3
(e)
i3
(h)
i=1
i=1
i=1
25
X
20
X
23
X
7i
(f)
i=1
(2i − 3)
i=1
(i)
i=1
(−i3 + 2i + 3)
(k)
(i2 − 8i − 9)
(i3 − 11i2 − 2)
i=1
2
(4i − 9i − 2)
(l)
11
X
(i3 −8i2 −i+12)
i=1
5. Sketch the region under the curve y = f (x) between x = a and x = b. Find its area by evaluating the
limit of an appropriate Riemann sum.
(a) y = 2x + 1 , a = 0 , b = 3
(f) y = −x2 + 4 , a = 0 , b = 2
(b) y = x + 3 , a = −1 , b = 1
(g) y = x2 + x + 1 , a = −1 , b = 2
(c) y = x2 , a = 0 , b = 2
(h) y = −x2 + x + 2 , a = −1 , b = 2
(d) y = x3 + 1 , a = 0 , b = 1
(i) y = −x3 + 2 , a = −1 , b = 1
(e) y = x2 + 2x + 1 , a = 0 , b = 3
(j) y = x3 + x2 , a = −1 , b = 1
6. Evaluate the following definite integrals using the definition as the limit of a Riemann sum.
Z
−6
Z
(x − 1) dx
(a)
−7
Z 2
Z
3
0
(−2x + 5) dx
Z
3
Z
3
4
(i)
(3x2 − x − 3) dx
−1
−3
−3
(x2 − 3x + 2) dx
0
(x − x ) dx
(f)
4
(h)
(2x + 1) dx
−1
Z 0
(c)
(2x2 + x − 1) dx
0
(e)
2
1
(g)
(3x − 1) dx
0
0
Z
Z
2
(d)
(x + 8) dx
(b)
1/2
Answers
√
√
√
√
2 + 5 + 8 + 11 + 14
1 1
1
1
1
1
(b) 1 − + −
+
−
+
2 5 10 17 26 37
1. (a)
2. (a)
√
11
X
2i − 1 =
i=5
(b)
5
X
i=1
10
X
2i + 1 =
i=4
6
X
(c) 12 + 22 + 32 + 42
(d) 1 + 3 + 32 + 33 + · · · + 3n
7
X
(c)
(−1)i+1 i2
9 + 2i
i=0
i=1
4
X
sin(i)
√
(d)
i
i
i=1
2i
(−1)
2i + 1
3. (a) n
i
(b)
n(n + 1)
2
(c)
n(n + 1)(2n + 1)
6
(d)
n(n + 1)
2
2
4. (a) 100
(c) 2275
(e) 23409
(g) 3645
(i) 14766
(k) −168
(b) 150
(d) 9455
(f) 360
(h) −5892
(j) 49
(l) 374
5. (a) 12
(b) 6
6. (a) −15/2
(c) 8/3
(e) 21
(g) 15/2
(i) 4
(d) 5/4
(f) 16/3
(h) 9/2
(j) 2/3
(d) −3/8
(g) 1/6
(b) 20
(e) 1/2
(h) 16/3
(c) 30
(f) 63/4
(i) 85/2