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The Definite Integral Quiz

Question

Web resources available

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.

Question 1

Which of the following is the left hand sum, with $n = 4,$ for $∫ 5 (1 + x2)dx? 1$

a)	34	b)	58
c)	60	d)	44.33

Your answer is correct.

Left hand sum $∑3 ∑3 f(xi)Δx = (1+ xi)Δx i=0 i=0$ where $Δx = 5--1-= 1, x0 = 1 4$ and $x4 = 5.$

So the left hand sum $= f(1)× 1+ f(2)× 1+ f(3)× 1 +f (4)× 1 = 2+ 5 +10 + 17 = 34$

Not correct. Choice (b) is false.

Try again, this is the right hand sum $= f(2)+ f(3)+ f(4)+ f(5) = 5+ 10 + 17 + 26 = 58$

Not correct. Choice (c) is false.

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.

Not correct. Choice (d) is false.

Try again, this is the exact answer for $∫ 5(1+ x2)dx. 1$

Question 2

We wish to find the right hand sum of $∫ 2 3 -3tdt$ with $n = 10.$
Which of the values of $ti$ below need to be substituted into $∑10 (ti)3Δt? i=1$

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)	$t1 = - 2.5, t1 = - 2, t2 = - 1.5,...,t10 = 2.$

Not correct. Choice (a) is false.

Try again, this is what you would require for a left hand sum.

Not correct. Choice (b) is false.

Try again, you need $Δt = 0.5.$

Not correct. Choice (c) is false.

Try again, you need $Δt = 0.5.$

Your answer is correct.

Since The right hand sum gives this approximation

we need to start the sum at $t1 = - 2.5$ and increase each term by $2 - (- 3) Δt = --------= 0.5 10$ to end at $t10 = 2.$

Question 3

Which of the following is the right hand sum, with $n = 6,$ for $∫ 2 (3x2 + 2x+ 1)dx? -1$

a)	11.625	b)	19.125
c)	20.125	d)	13

Not correct. Choice (a) is false.

Try again, this is the left hand sum.

Your answer is correct.

Right hand sum $∑6 ∑6 2 f(xi)Δx = (3xi + 2xi + 1)Δx i=1 i=1$ where $2-+-1 Δx = 6 = 0.5, x0 = - 1$ and $x = 2. 6$

So the right hand sum
$= 0.5(f (- 0.5)+ f(0)+ f(0.5)+ f(1) +f (1.5)+ f(2))$ $= 0.5(0.75+ 1 +2.75+ 6 +10.75+ 17) = 19.125$

Not correct. Choice (c) is false.

Try again, there are too many terms

Not correct. Choice (d) is false.

Try again, this is the exact value of $∫ 2 2 (3x + 2x+ 1)dx. -1$

Question 4

Which of the following would be the appropriate integral to evaluate if you were asked to find the area between the curve $4 y = 16 - x$ and the $x$ -axis?

a)	$∫ 2 (16- x4)dx -2$	b)	$∫ 4 (16- x4)dx -4$
c)	$∫ -2(16- x4)dx 2$	d)	$∫ 16 4 0 (16- x )dx$

Your answer 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.