MATH 141 webworks .Set1
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8/9/2019 MATH 141 webworks .Set1
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Razan Khrais Assignment Set1 MATH 141, Winter 2015due 01/26/2015 at 11:55pm EST.
You may attempt any problem an unlimited number of times.
1. (1 pt) Consider the integral
Z 115
(3 x2 + 4 x + 1) dx
(a) Find the Riemann sum for this integral using right end-
points and n = 3. R3 =(b) Find the Riemann sum for this same integral, using left end-
points and n = 3. L3 =
Correct Answers:
• 1728• 1104
2. (1 pt) The following sum 16− 4
n
2 · 4n +
16− 8
n
2 · 4n + . . .+
16− 4n
n
2 · 4n
is a right Riemann sum for the definite integralZ b
0 f ( x) dx
where b =
and f ( x) =The limit of these Riemann sums as n → ∞ isCorrect Answers:
• 4• sqrt(16 - xˆ2)• 12.566370616
3. (1 pt) Consider the function f ( x) = − x2
2 + 6.
In this problem you will calculate
Z 30
− x
2
2 + 6
dx by us-
ing the definition
Z ba
f ( x) dx = limn→∞
n
∑i=1
f ( xi)∆ x
The summation inside the brackets is Rn which is the Rie-
mann sum where the sample points are chosen to be the right-
hand endpoints of each sub-interval.
Calculate Rn for f ( x) =−
x2
2 + 6 on the interval [0,3] and
write your answer as a function of n without any summation
signs.
Rn =lim
n→∞ Rn =
Correct Answers:
• 6*3 + 3**3*(n+1)*(2*n+1)/(6*(-2)*n**2)
• 13.5
4. (1 pt) On a sketch of y = e x, represent the left Riemann
sum with n = 2 approximatingR
21 e
x dx. Write out the terms of
the sum, but do not evaluate it:
Sum = +
On another sketch, represent the right Riemann sum with
n = 2 approximatingR 2
1 e x dx. Write out the terms of the sum,
but do not evaluate it:
Sum = +
Which sum is an overestimate?
• A. the right Riemann sum• B. the left Riemann sum• C. neither sum
Which sum is an underestimate?
• A. the left Riemann sum• B. the right Riemann sum• C. neither sum
SOLUTION
Sketches of the left and right Riemann sums are shown be-
low. The region we’re integrating over is 1 ≤ x ≤ 2, which iswhere the function is drawn with a heavy line.
left sum right sum
Thus we can see that the left Riemann sum is given by
Z 21
e x dx ≈ e1 ·0.5 + e1.5 ·0.5
and the right Riemann sum by
Z 21
e x dx ≈ e1.5 ·0.5 + e2 ·0.5.
From the sketches we can see that the right Riemann sum is
an overestimate and the left Riemann sum an underestimate.Correct Answers:
• eˆ1*0.5• eˆ1.5*0.5• eˆ1.5*0.5• eˆ2*0.5• A• A
1
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8/9/2019 MATH 141 webworks .Set1
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5. (1 pt) Express the expression 1
n
n
∑k =1
cos
6k π
19n
as the
Riemann sum for an integral of the form
Z 6π/190
f (t )dt for a
suitable function f .
Hence find limn→∞
1
n
n
∑k =1
cos
6k π
19n
= .
Correct Answers:
• 0.8438481603066636. (1 pt) Use part I of the Fundamental Theorem of Calculus
to find the derivative of
f ( x) =
Z x−1
t 3 + 1dt
f ( x) =NOTE: Enter a function as your answer. Make sure that your
syntax is correct, and the variable is x.Correct Answers:
• sqrt(xˆ3+1)7. (1 pt) Use part I of the Fundamental Theorem of Calculus
to find the derivative of
h( x) =
Z sin( x)−1
(cos(t 5) + t ) dt
h( x) =
Correct Answers:
• cos(x)*(cos((sin(x))ˆ5)+sin(x))
8. (1 pt) Evaluate the integral below by interpreting it interms of areas. In other words, draw a picture of the region the
integral represents, and find the area using high school geome-
try.Z 2−2
4− x2dx =
Correct Answers:
• 6.2831853089. (1 pt) Evaluate the integral by interpreting it in terms of
areas. In other words, draw a picture of the region the integral
represents, and find the area using high school geometry.Z 30|6 x−8|dx =
Correct Answers:• 13.6666666666667
10. (1 pt) Evaluate the integral
Z √ 31
7
1 + x2dx
Correct Answers:
• 1.8325957145940511. (1 pt) Evaluate the definite integral
Z 63
8 x2 + 3√ x
dx
Correct Answers:
• 236.60278872192412. (1 pt) Evaluate the definite integralZ π
08sin( x)dx
Correct Answers:• 1613. (1 pt) Let
f ( x) =
0 if x