# Definite Integral Quiz – Quiz On Simple Integrals

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This quiz tests the work covered in Lecture 21 and corresponds to Section 5.2 of thetextbook Calculus: Single and Multivariable (Hughes-Hallett, Gleason, McCallum etal.).There are no Wiley web quizzes for this topic.

Đang xem: Integral quiz

If you need some help on sigma notation then go tohttp://quiz.econ.wnyrails.org.edu.au/mathquiz/sigma/index.php for quizzes and webresources.

Questions 6 and 7 at http://www.wnyrails.org/u/UG/JM/MATH1011/Quizzes/quiz10.htmlcould be useful.

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

Which of the following is the left hand sum, withn = 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(xi)Δx = ∑ i=03(1 + xi)Δx whereΔx = 5 − 1 4 = 1,x0 = 1 andx4 = 5.

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
Tryagain, 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 = 60which 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 withn = 10.Which of the values of ti below needto be substituted into ∑ i=110(ti)3Δt? Exactlyone 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!
SinceThe right hand sum gives this approximation

we need to start the sum at t1 = −2.5and increase each term by Δt = 2 − (−3) 10 = 0.5to end at t10 = 2.
Which of the following is the right hand sum, withn = 6, for∫ −12(3×2 + 2x + 1)dx? Exactlyone 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 handsum ∑ i=16f(xi)Δx = ∑ i=16(3xi2 + 2xi + 1)Δxwhere Δx = 2 + 1 6 = 0.5,x0 = −1and x6 = 2.

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(3×2 + 2x + 1)dx.
Which of the following would be the appropriate integral toevaluate if you were asked to find the area between the curvey = 16 − x4 and thex-axis? Exactly oneoption must be correct)

a) ∫ −22(16 − x4)dx
b) ∫ −44(16 − x4)dx
c) ∫ 2−2(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-axisat x = −2 andx = 2 so this is the most appropriateintegral.
Choice (b) is incorrect
Try again, you needto factorize 16 − x4 to determinewhere 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 tofactorize 16 − x4 to determinewhere it cuts the x-axis.