III - The Definite Integral - SLC Home Page

MATHEMATICS 201-203-RE
Integral Calculus
Martin Huard
Winter 2009
III - The Definite Integral
1. The graph of a function f is given.
8
Estimate ∫ f ( x ) dx using four subintervals with
0
a) right endpoints
b) left endpoints
c) midpoints
4
2
y
2 4 6
8
x
2. The graph of a function f is given.
11
Estimate f ( )
∫ x dx using five
1
subintervals with
a) right endpoints
b) left endpoints
c) midpoints
y
4
3
2
1
-1
-2
1 2 3 4 5 6 7 8 9 10 11 x
3. Evaluate each of the following definite integral using a Riemann Sum. (The answers are
given using the RHE). Does the answer represent an area? Explain.
2
1
a) ( − )
∫
0
1 x dx
3
c) ( + x )( − )
∫
−
2
1 1 3
x dx
4
e) ∫ ( 2
0
2
+ −1)
3
g) ∫ ( −
−1
2
+ 4 − 5)
1
3
i) ∫ ( x + 1)
dx
−1
x x dx
x x dx
1
b) ( x + )
∫
−
2 2 1
dx
3
2
d) ∫ ( − 4 + 3)
0
x x dx
−1
f) ( 4 − 3)( 3 + 1)
∫
−3
x x dx
5
2
h) ∫ ( + + 1)
∫
2
x x dx
3 2
j) ( + 2)( −1)
−3
x x dx
4. Prove the following.
2 2
b b − a
a) ∫ x dx = . b)
a 2
3 3
b
2 b − a
∫ x dx = .
a 3

Math 203
III – The Definite Integral
6
6
3
5. If ∫ f ( t)
dt = 2 and ∫ f ( t)
dt = 5, find ( )
1
3
∫ f t dt .
1
4
1
4
6. If ∫ f ( x)
dx = 7 and ∫ f ( x)
dx = −1, find ( )
−1
−1
∫ 5 f x dx .
1
7. Use the properties of integrals to verify the inequality without evaluating the integral.
π
π
2 2
12
≤ ∫ sin x dx ≤
π
3
6
π
8. Use the properties of integrals (along with question 3) to prove
9. Evaluate the integral by interpreting it in terms of area.
2
a) ∫ f ( x ) dx
−2
5
b) ∫ f ( x ) dx
2
4
c) ∫ f ( x ) dx
0
5
d) ∫ f ( x ) dx
−2
π
2
2
π
∫ xsin
x dx ≤ .
0
8
y = f ( x)
10. Express the limit as a definite integral.
a)
lim
n
i
6
∑ b)
n→∞ 7
i = 1 n
n πi
π sin
2n
c) lim∑ d)
n→∞ = 2n
i 1
n
3i
3 2 +
n
lim∑
n→∞ i = 1 n
n 2 2
n − i
lim∑
n→∞ 2
i = 1 n
Winter 2009 Martin Huard 2

Math 203
III – The Definite Integral
Answers
1. a) 18 b) 20 c) 22
2. a) 8 b) 14 c) 6
( n −
3. a) lim 1)
9
= 1 Yes b) lim = 0
n→∞
n
n→∞
n
No
32( n −1)(
n + 1) 32
− 9( n −1)
c) lim = Yes d) lim = 0
n→∞
2 2
3n
3
n→∞
2n
No
2
4(7n
+ 8)(5n
+ 2) 140
2(59n
− 53n
+ 8)
e) lim = No f) lim = 118
n→∞
2 2
3n
3
n→∞
n
Yes
2
2
−8(5n
− 6n
+ 4) 40
3(35n
+ 24n
+ 3) 105
g) lim
= − No h) lim
= Yes
n→∞
2
2
3n
3
n→∞
2n
2
2( n +
i) lim 1)
12( n + 9)
= 2 Yes j) lim = 12 No
n→∞
n
n→∞
n
4. Use the Riemann sum
5. –3
6. 40
7. Use the inequality 1 π π
2
≤ sin x ≤ 1 for x ∈⎡⎣ 6
,
2
⎤⎦ π
8. Use the inequality xsin
x ≤ x for x ∈⎡⎣
0,
2
⎤⎦
9. a) 5 b) -2 c) 3 2 d) 3
10. a)
1
x 6
dx
0
∫ b)
5
∫ xdx c)
2
π
2
∫ sin x dx d)
0
∫
1
0
2
1−
x dx
Winter 2009 Martin Huard 3