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Quiz 11: One variable integration

Last unanswered question  Question  Next unanswered question
 

Question 1

 
 
Evaluate ∫ 1 (6x2 - 4x+ 3) dx 0 .
a) 4   b) -3
c) 3   d) 5

 

Not correct. Choice (a) is false.
Not correct. Choice (b) is false.
Your answer is correct.
∫ 1(6x2 - 4x+ 3) dx = [2x3 - 2x2 + 3x ]1 = 2- 2+ 3 = 3. 0 0
Not correct. Choice (d) is false.
 

Question 2

 
 
Evaluate ∫ π- 2 (sinx + cosx ) dx 0 .
a) 2   b) 0
c) π- 2   d) -2

 

Your answer is correct.
∫ π 2(sin x+ cosx) dx = [- cosx + sinx]π2 0 0 = - cos π-+ sin π-- (- cos 0+ sin0) 2 2 = 1+ 1 = 2.
Not correct. Choice (b) is false.
Not correct. Choice (c) is false.
Not correct. Choice (d) is false.
 

Question 3

 
 
Evaluate π ∫ 4- sin 2x dx 0 .
a) 1   b) 1 - 2
c) 1 - - 2   d) -1

 

Not correct. Choice (a) is false.
Your answer is correct.
∫ π4 [ ]π4 sin2x dx = - 1cos2x 0 2 0 = - 1cos π-+ 1 cos0 2 2 2 = 1. 2
Not correct. Choice (c) is false.
Not correct. Choice (d) is false.
 

Question 4

 
 
Evaluate ∫ √3 3 x (1+ x2)2 dx 0 .
a) 32 5   b) 31 5
c) 3 2   d) 9 2

 

Not correct. Choice (a) is false.
Your answer is correct.
√- √- ∫ 3 ( 2) 32 [1 2 5] 3 0 x 1 + x dx = 5(1+ x )2 1 5 1 0 = - (4)2 - - 532 1 531 = -- - - = --. 5 5 5
Not correct. Choice (c) is false.
Not correct. Choice (d) is false.
 

Question 5

 
 
Evaluate ∫ 2 3 3x2ex dx 0 .
a) e8 - 1   b) 8e8
c) - e8   d) 8 12e - 12

 

Your answer is correct.
∫ 2 2 x3 [ x3]2 8 0 8 0 3x e dx = e 0 = e - e = e - 1.
Not correct. Choice (b) is false.
Not correct. Choice (c) is false.
Not correct. Choice (d) is false.
 

Question 6

 
 
Let A be the area under the curve f(x) = 16 - x2 on the interval [0,4]. Dividing the interval into 4 subintervals and defining xi to be the midpoint of the ith interval (the area of the rectangle is thus f(xi) times the length of the interval), the best estimate for A is
a) ∑4 2 1 A = (16- xi)× 2 i=1 , where i--1 xi = 2
b) ∑4 A = (16- x2i)× 1 i=1 2 , where xi = i-- 1 2
c) ∑4 2 A = (16- xi) i=1 , where 2i--1 xi = 2
d) ∑3 A = (16- x2i) i=0 , where 2i- 1 xi = ----- 2

 

Not correct. Choice (a) is false.
Not correct. Choice (b) is false.
Your answer is correct.
The intervals are of length 1, and the area is calculated at 1 2, 3 2, 5 2 and 7 2.
Not correct. Choice (d) is false.
 

Question 7

 
 
Which of the following sums is the best estimate for A, the area under the curve f(x) = x3 + 2 on the interval [-1,2], divided into 6 subintervals and choosing xi as the right-endpoint of the ith interval ?
a) 6 A = ∑ (x3+ 2) i=0 i , where xi = i--3 2
b) ∑6 3 1 A = (xi + 2)× 2 i=0 , where 2i - 1 xi =--2--
c) 6 A = ∑ (x3+ 2) i=1 i , where xi = 2i--3 2
d) ∑6 3 1 A = (xi + 2)× 2 i=1 , where i - 2 xi =--2-

 

Not correct. Choice (a) is false.
Not correct. Choice (b) is false.
Not correct. Choice (c) is false.
Your answer is correct.
The intervals are of length 1 2, and the area is calculated at -1 2, 0, 1 2, 1, 3 2, 2
 

Question 8

 
 
Find the indefinite integral ∫ 2t2(1+ t3)4 dt.
a) 1 -(1+ t3)5 + C 5   b) 2 3 5 5 (1 + t) + C
c) 2 t3(1+ t3)5 + C 3   d) -2 3 5 15 (1 + t) + C

 

Not correct. Choice (a) is false.
Not correct. Choice (b) is false.
Not correct. Choice (c) is false.
Your answer is correct.
Let u = 1 + t3, du = 3t2 dt .
∫ 2 34 2 ∫ 4 2t(1+ t )dt = 3 u du 2 = -- u5 + C 152 = -- (1 + t3)5 + C. 15
 

Question 9

 
 
Find the indefinite integral ∫ sin3xcosx dx.
a) sin4x + C   b) 1 4 sin4x + C
c) 3sin2x + C   d) 1 4 - 4 sin x+ C

 

Not correct. Choice (a) is false.
Your answer is correct.
Let u = sinx, du = cosx dx.
∫ ∫ sin3 xcosx dx = u3 du = 1u4 + C 4 = 1 sin4x + C. 4
Not correct. Choice (c) is false.
Not correct. Choice (d) is false.
 

Question 10

 
 
Find the indefinite integral
∫ ---2t--e-t +-4---dt. (t2 + 4t+ e-t + 1)2
a) -------1------ t2 + 4t+ e-t + 1 + C   b) --------- 1-------+ C 3(t2 + 4t+ e-t + 1)3
c) -2------1-t--- + C t + 4t+ e + 1   d) ---------1--------+ C 3(t2 + 4t+ e-t + 1)3

 

Not correct. Choice (a) is false.
Not correct. Choice (b) is false.
Your answer is correct.
Let u = t2 + 4t+ e-t + 1 , du = (2t+ 4- e-t) dt.
∫ -t ∫ ---2t--e--+-4----dt = -1 du (t2 + 4t+ e-t + 1)2 u2 = - u-1 + C ------- 1----- = t2 + 4t+ e-t + 1 + C.
Not correct. Choice (d) is false.