Chapter 5 | Integration
547
130. Suppose ⎡
⎤ ⎦ can be subdivided into subintervals
⎣ a , b
121. [T] y = xe x 2
over the interval [0, 2]; the exact
a = a 0 < a 1 < a 2 <⋯< a N = b such that either f ≥0 over [ a i −1 , a i ] or f ≤0 over [ a i −1 , a i ]. Set A i = ∫ a i −1 a i f ( t ) dt . a. Explain why ∫ a b f ( t ) dt = A 1 + A 2 +⋯+ A N . b. Then, explain why | ∫ a b f ( t ) dt | ≤ ∫ a b | f ( t ) | dt . 131. Suppose f and g are continuous functions such that ∫ c d f ( t ) dt ≤ ∫ c d g ( t ) dt for every subinterval ⎡ ⎣ c , d ⎤ ⎦ of ⎡ ⎣ a , b ⎤ ⎦ . Explain why f ( x ) ≤ g ( x ) for all values of x . 132. Suppose the average value of f over ⎡ ⎣ a , b ⎤ ⎦ is 1 and the average value of f over ⎡ ⎣ b , c ⎤ ⎦ is 1 where a < c < b . Show that the average value of f over [ a , c ] is also 1. 133. Suppose that ⎡ ⎣ a , b ⎤ ⎦ can be partitioned. taking a = a 0 < a 1 <⋯< a N = b such that the average value of f over each subinterval [ a i −1 , a i ] =1 is equal to 1 for each i =1,…, N . Explain why the average value of f over ⎡ ⎣ a , b ⎤ ⎦ is also equal to 1. 134. Suppose that for each i such that 1≤ i ≤ N one has ∫ i −1 i f ( t ) dt = i . Show that ∫ 0 N f ( t ) dt = N ( N +1) 2 . 135. Suppose that for each i such that 1≤ i ≤ N one has ∫ i −1 i f ( t ) dt = i 2 . Show that ∫ 0 N f ( t ) dt = N ( N +1)(2 N +1) 6 . 136. [T] Compute the left and right Riemann sums L 10 and R 10 and their average L 10 + R 10 2 for f ( t ) = t 2 over [0, 1]. Given that ∫ 0 1 t 2 dt =0.33 – , to how many decimal places is L 10 + R 10 2 accurate?
solution is 1 4 ⎛
⎞ ⎠ .
⎝ e 4 −1
⎛ ⎝ 1 2
⎞ ⎠
x over the interval [0, 4]; the exact
122. [T] y =
15 64ln(2)
.
solution is
123. [T] y = x sin ⎛
⎞ ⎠ over the interval [− π , 0]; the
⎝ x 2
⎛ ⎝ π 2
⎞ ⎠ −1
cos
.
exact solution is
2 π
2 π
A = ∫
sin 2 tdt
124.
Suppose that
and
0
2 π cos 2 tdt . Show that A + B =2 π and A = B .
B = ∫
0
− π /4 π /4
A = ∫
sec 2 tdt = π
125.
Suppose that
and
− π /4 π /4
tan 2 tdt . Show that A − B = π 2 .
B = ∫
126. Show that the average value of sin 2 t over [0, 2 π ] is equal to 1/2 Without further calculation, determine whether the average value of sin 2 t over [0, π ] is also equal to 1/2. 127. Show that the average value of cos 2 t over [0, 2 π ] is equal to 1/2. Without further calculation, determine whether the average value of cos 2 ( t ) over [0, π ] is also equal to 1/2. 128. Explain why the graphs of a quadratic function (parabola) p ( x ) and a linear function ℓ ( x ) can intersect in at most two points. Suppose that p ( a ) = ℓ ( a ) and p ( b ) = ℓ ( b ), and that ∫ a b p ( t ) dt > ∫ a b ℓ ( t ) dt . Explain why ∫ c d p ( t ) > ∫ c d ℓ ( t ) dt whenever a ≤ c < d ≤ b . 129. Suppose that parabola p ( x ) = ax 2 + bx + c opens downward ( a <0) and has a vertex of y = − b 2 a >0. For which interval [ A , B ] is ∫ A B ⎛ ⎝ ax 2 + bx + c ⎞ ⎠ dx as large as possible?
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