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Quiz 3: Areas, Volumes and Integration by Substitution

Last unanswered question  Question  Next unanswered question
 

Question 1

 
 
What is the area bounded by the graphs of f(x) = sin x and g(x) = cosx and the lines x = 0 and x = π- 2  ?
a) 2(√2 - 1)   b) 2
c) √ - 2 2   d) √- 1 + 2 2
e) None of the above

 

Your answer is correct.
The area is given by ∫π∕4 2 0 (cosx- sinx) dx and this equals √- 2( 2 - 1).
Not correct. Choice (b) is false.
Have you sketched the graphs and found their point of intersection? Try to use symmetry to make your work easier.
Not correct. Choice (c) is false.
Have you sketched the graphs and found their point of intersection? Try to use symmetry to make your work easier.
Not correct. Choice (d) is false.
Have you sketched the graphs and found their point of intersection? Try to use symmetry to make your work easier.
Not correct. Choice (e) is false.
 

Question 2

 
 
What is the area bounded by the graphs of √----- f(x) = x - 1 , 2 g(x) = -- x , the x - axis and the line x = 3  ? Enter your answer correct to 2 decimal places. Do not enter any units.

 

Your answer is correct
Well done. The required area is ∫ 2√ ----- ∫ 3 2- 1 x- 1 dx+ 2 x dx and this equals 2 + 2ln 3 ≈ 1.48 3 2 .
Not correct. You may try again.
Sketch the curves and find their point of intersection. Now express the required area as the sum of two separate integrals.
 

Question 3

 
 
Which option below is an integral which equals the volume of the solid of revolution formed when the area between the curves y = √4x- and y = 2x3 and between the lines x = 0 and x = 1 is rotated about the line x = 3  ?
a) ∫ 22πx(√4x-- 2x3) dx 0   b) ∫1 4π(3- x)(√x--- x3) dx 0
c) ∫2 √-- 3 0 2π(3- x)( x - x ) dx   d) ∫ 1 3 √ -- 0 2π(3 - x)(x - x) dx

 

Not correct. Choice (a) is false.
Which method are you thinking about - shells or discs? Check that you have the right expression to integrate and the right endpoints on the integral sign.
Your answer is correct.
This is the integral which gives the required volume when the method of shells is used. The radius of the shell is 3- x and the height √4x--- 2x3 . Values of x range from 0 to 1.
Not correct. Choice (c) is false.
Check the limits on the integral sign.
Not correct. Choice (d) is false.
Which is greater on the interval concerned, √ --- 4x or 3 2x  ?
 

Question 4

 
 
The method of shells is used to obtain the volume V of the solid of revolution formed when the area between the curve 2 y = x and the x -axis, from x = 0 to x = 1 , is rotated about the line y = - 2 . This method gives us
∫ 1 √ -- V = 2π 0 (2 + y)(1- y) dy.
Which of the integrals below is the one which calculates the same volume by the method of discs?
a) ∫1 2 0 π((y + 2) - 2) dy   b) ∫ 1 2 0 π((x + 2)- 4) dx
c) ∫ 1 0 π((x2 + 2)2 - 4) dx   d) ∫1 0 π((x2 + 2)2 + 4) dx
e) ∫1 0 π((x2 + 2)- 1) dx

 

Not correct. Choice (a) is false.
Not correct. Choice (b) is false.
Your answer is correct.
Not correct. Choice (d) is false.
Not correct. Choice (e) is false.
 

Question 5

 
 
Use a substitution to find ∫ t √-----4 dt. 4- t
a) 2 2 sinh(t2-)+ C   b) 1 sin(2 - t2)+ C 2
c) - 1 ln(2- t2)+ C 2   d) -1 t2- cos (2 )+ C
e) 1 sin-1(t22 )+ C 2

 

Not correct. Choice (a) is false.
You can check if your answer is right by differentiating it. Do this and you will see why this option is not correct.
Not correct. Choice (b) is false.
You can check if your answer is right by differentiating it. Do this and you will see why this option is not correct.
Not correct. Choice (c) is false.
It is not valid to simplify √----4 4 - t to √ - √-4 4- t .
Not correct. Choice (d) is false.
You can check if your answer is right by differentiating it. Do this and you will see why this option is not correct.
Your answer is correct.
Useful substitutions are u = t2 or 2 u = t- 2 .
 

Question 6

 
 
Use a substitution to find ∫ √ ----- tanx sec4 x dx. Hint: try a substitution that gets rid of the square root sign.
a) -2(tanx)32 sec5x + C 15   b) 3 (tanx) 32 + 2(tanx)92 + C 2 9
c) 2 3 2 7 3 (tanx) 2 + 7(tanx)2 + C   d) 2sec6 x√tanx-+ C
e) None of the above

 

Not correct. Choice (a) is false.
Differentiate your answer to see why this is incorrect. Try the substitution √----- u = tanx .
Not correct. Choice (b) is false.
Differentiate your answer to see why this is incorrect. Try the substitution √----- u = tanx .
Your answer is correct.
The substitution u = √tan-x , together with the identity 1 + tan2 x = sec2x are useful in this problem.
Not correct. Choice (d) is false.
Differentiate your answer to see why this is incorrect. Try the substitution u = √tanx- .
Not correct. Choice (e) is false.
 

Question 7

 
 
Use the shell method to find the volume of the solid formed when the area enclosed by the curves y = x2 - 4x+ 3 and y = - x2 + 2x+ 3 is rotated about the y -axis. Enter your answer correct to two decimal places. (Do not enter any units.)

 

Your answer is correct
Well done! Your integral should be 2π∫ 3 x(- 2x2 + 6x) dx = 27π, 0 which is 84.82 correct to two decimal places.
Not correct. You may try again.
To use the shell method, you need the radius (in this case, just x ) and the height of the shell (in this case, (- x2 + 2x+ 3)- (x2 - 4x + 3) ). Now put this information together to obtain the volume of the shell, and then integrate.
 

Question 8

 
 
Use the disc method to find the volume of the solid formed when the area enclosed by the curvey = sin x and the x -axis, between x = 0 and x = π , is rotated about the line y = 2 . Enter your answer correct to two decimal places. (Do not enter any units.)

 

Your answer is correct
Not correct. You may try again.
The disc method requires us to set up an integral describing a disc with a hole, formed by rotating the area about the line y = 2 . The outer radius of the disc is 2 and the inner radius is 2- sin x .
 

Question 9

 
 
Find the volume of the solid formed when the area enclosed by the curvesy = ex2 and y = ex , and the line x = 0 , is rotated about the y -axis. You may use the fact that ∫ xex dx = ex(x - 1) + C.
a) 2π(1 - e4)   b) 2π e2- 4
c) 2π(2e- 1) 3   d) πe4
e) None of the above

 

Your answer is correct.
The volume is given by the integral 2π ∫1 x(ex - ex2) dx. 0 When evaluated using the hint in the question, this equals 2π(1- e 4 .
Not correct. Choice (b) is false.
Not correct. Choice (c) is false.
Not correct. Choice (d) is false.
Not correct. Choice (e) is false.
 

Question 10

 
 
What is the volume of the solid obtained when the area between the x -axis and the curve 3 y = x +1 , from x = - 1 to x = 1 , is rotated about the line y = - 1  ?
a) 20π ---- 3   b) 10π- 3
c) 21π- 4   d) 4π- 1
e) None of the above

 

Not correct. Choice (a) is false.
Hint: the disc method is easier here than the shell method. The inner radius is 1. What is the outer radius?
Not correct. Choice (b) is false.
Hint: the disc method is easier here than the shell method. The inner radius is 1. What is the outer radius?
Not correct. Choice (c) is false.
Hint: the disc method is easier here than the shell method. The inner radius is 1. What is the outer radius?
Not correct. Choice (d) is false.
Hint: the disc method is easier here than the shell method. The inner radius is 1. What is the outer radius?
Your answer is correct.
The correct answer is ∫ 1 44π π -1((x3 + 2)2 - 1) dx =-7-. This integral is set up using the disc method. Using the shell method, the integral is 2π ∫02(y + 1)(1 - (y - 1)13) dy , a more difficult integral to evaluate.