Quiz 11
1. A table of values of an increasing function f is shown. Find lower and upper estimates for
30
∫10 f (x) dx.
x
10 14 18 22 26 30
f (x) -12 -6 -2
1
3
8
Solution: ∆x = 4, hence
L5 = 4(−12 − 6 − 2 + 1 + 3) = −64,
R5 = 4(−6 − 2 + 1 + 3 + 8) = 16.
The function being increasing, the Riemann sum with left endpoint L5 = −64 is a lower estimate,
while the sum with right endpoints R5 = 16 is an upper estimate.
√
8
2. Use the Midpoint Rule with the given values of n = 4 to approximate the integral ∫0 sin x dx.
√
√
√
√
√
√
(Use
the appropriate calculations:
0+sin 2+sin
4+sin 6 ≈ 2.535, sin 1+sin 3+
√ one of √
√
√ sin √
√
sin 5 + sin 7 ≈ 3.091, sin 2 + sin 4 + sin 6 + sin 8 ≈ 2.8432. )
Solution: ∆x = 8−0
, hence the nodes are x0 = 0, x1 = 2, x2 = 4, x3 = 6, x4 = 8, respectively the
4
midpoints x∗1 = 1, x∗2 = 3, x∗3 = 5, x∗4 = 7. Therefore
8
∫
sin
√
√
x dx ≈ 2( sin
√
1 + sin
3 + sin
√
√
5 + sin 7) ≈ 2 ⋅ 3.091 = 6.182.
0
10 1
x
3. Use the properties of integrals to verify: .9 ≤ ∫1
dx ≤ 9.
Solution:
Since
1
10
≤
1
x
10 1
10
≤ 1, ∀x ∈ [1, 10], we have ∫1
10 1
x
dx ≤ ∫1
10
dx ≤ ∫1 1 dx
⇒
10 1
x
.9 ≤ ∫1
dx ≤ 9.
4. The graph of f is shown. Evaluate each integral by interpreting it in terms of areas.
2
(a) ∫0 f (x) dx =
2⋅(1+3)
2
= 4.
5
3
5
7
5
7
(b) ∫2 f (x) dx = ∫2 f (x) dx + ∫3 f (x) dx = 1 ⋅ 3 +
(c) ∫3 f (x) dx = ∫3 f (x) dx + ∫5 f (x) dx =
(5−3)⋅3
2
(5−3)⋅3
2
+
= 3 + 3 = 6.
(7−5)⋅(−3)
2
= 3 − 3 = 0.