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MATH1111 Quizzes

The Definite Integral Quiz
Web resources available Questions

This quiz tests the work covered in Lecture 21 and corresponds to Section 5.2 of the textbook Calculus: Single and Multivariable (Hughes-Hallett, Gleason, McCallum et al.).

There are no Wiley web quizzes for this topic.

If you need some help on sigma notation then go to http://quiz.econ.usyd.edu.au/mathquiz/sigma/index.php for quizzes and web resources.

Questions 6 and 7 at http://www.maths.usyd.edu.au/u/UG/JM/MATH1011/Quizzes/quiz10.html could be useful.

There is an excellent animation of the general Riemann sum at http://archives.math.utk.edu/visual.calculus/4/riemann_sums.4/index.html.

Which of the following is the left hand sum, with n = 4, for 15(1 + x2)dx? Exactly one option must be correct)
a)
34
b)
58
c)
60
d)
44.33

Choice (a) is correct!
Left hand sum i=03f(x i)Δx = i=03(1 + x i)Δx where Δx = 5 1 4 = 1,x0 = 1 and x4 = 5.
PIC
So the left hand sum = f(1) × 1 + f(2) × 1 + f(3) × 1 + f(4) × 1 = 2 + 5 + 10 + 17 = 34
Choice (b) is incorrect
Try again, this is the right hand sum = f(2) + f(3) + f(4) + f(5) = 5 + 10 + 17 + 26 = 58
Choice (c) is incorrect
Try again, this is = f(1) + f(2) + f(3) + f(4) + f(5) = 2 + 5 + 10 + 17 + 26 = 60 which has one term too many.
Choice (d) is incorrect
Try again, this is the exact answer for 15(1 + x2)dx.
We wish to find the right hand sum of 32t3dt with n = 10.
Which of the values of ti below need to be substituted into i=110(t i)3Δt? Exactly one option must be correct)
a)
t0 = 3,t1 = 2.5,t2 = 2,,t9 = 2.
b)
t0 = 3,t1 = 2,t1 = 1,,t9 = 1.
c)
t1 = 2,t2 = 1,t3 = 0,,t10 = 2.
d)
t1 = 2.5,t1 = 2,t2 = 1.5,,t10 = 2.

Choice (a) is incorrect
Try again, this is what you would require for a left hand sum.
Choice (b) is incorrect
Try again, you need Δt = 0.5.
Choice (c) is incorrect
Try again, you need Δt = 0.5.
Choice (d) is correct!
Since The right hand sum gives this approximation
PIC
we need to start the sum at t1 = 2.5 and increase each term by Δt = 2 (3) 10 = 0.5 to end at t10 = 2.
Which of the following is the right hand sum, with n = 6, for 12(3x2 + 2x + 1)dx? Exactly one option must be correct)
a)
11.625
b)
19.125
c)
20.125
d)
13

Choice (a) is incorrect
Try again, this is the left hand sum.
Choice (b) is correct!
Right hand sum i=16f(x i)Δx = i=16(3x i2 + 2x i + 1)Δx where Δx = 2 + 1 6 = 0.5,x0 = 1 and x6 = 2.
PIC
So the right hand sum
= 0.5(f(0.5) + f(0) + f(0.5) + f(1) + f(1.5) + f(2)) = 0.5(0.75 + 1 + 2.75 + 6 + 10.75 + 17) = 19.125
Choice (c) is incorrect
Try again, there are too many terms
Choice (d) is incorrect
Try again, this is the exact value of 12(3x2 + 2x + 1)dx.
Which of the following would be the appropriate integral to evaluate if you were asked to find the area between the curve y = 16 x4 and the x-axis? Exactly one option must be correct)
a)
22(16 x4)dx
b)
44(16 x4)dx
c)
22(16 x4)dx
d)
016(16 x4)dx

Choice (a) is correct!
The curve y = 16 x4 = (x2 4)(x2 + 4) = (x 2)(x + 2)(x2 + 4) cuts the x-axis at x = 2 and x = 2 so this is the most appropriate integral.
Choice (b) is incorrect
Try again, you need to factorize 16 x4 to determine where it cuts the x-axis.
Choice (c) is incorrect
Try again, this will give you a negative value for the area.
Choice (d) is incorrect
Try again, you need to factorize 16 x4 to determine where it cuts the x-axis.