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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

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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