Docsity
Docsity

Prepare for your exams
Prepare for your exams

Study with the several resources on Docsity


Earn points to download
Earn points to download

Earn points by helping other students or get them with a premium plan


Guidelines and tips
Guidelines and tips

Math 106 A - Exam 01: Integration and Volume Calculation, Exams of Calculus

A math exam focusing on integration and volume calculation. Students are required to set up and evaluate integrals representing the area and arc length of regions between parabolas and straight lines, as well as the volumes of solids of revolution. They are also asked to estimate errors using given formulas and the values from a table of velocities.

Typology: Exams

2012/2013

Uploaded on 03/16/2013

parni
parni 🇮🇳

4.1

(14)

102 documents

1 / 5

Toggle sidebar

This page cannot be seen from the preview

Don't miss anything!

bg1
Math 106 A Exam 01 page 1 02/01/2013 Name
1. Consider the region Sin the plane of all points which are be-
tween the graphs of the parabola y= 9 x2and the straight line
y=x+ 3. The region Sis shown in the figure to the right.
1A. Set up the integral which represents the area of Sif the corre-
sponding approximations are rectangles each of whose base width
is 4xand each rectangle goes from a top curve down to a bottom
curve. That is, the integral is of the form Rdx.
1B. Evaluate the integral in (1A) to find that area. Show all your work.
2A. Set up the integral which gives the arc length of the curved part of the boundary of S, that is, the arc length of the
graph of y= 9 x2from x=3 to x= 2.
2B. The integral in 2A is “doable” the back of your book has a formula for the antiderivative you’d need. But it’s
so complicated, and in practice, a good numerical approximation will do. Indeed, find the MID(50) approximation for the
integral in 2A.
pf3
pf4
pf5

Partial preview of the text

Download Math 106 A - Exam 01: Integration and Volume Calculation and more Exams Calculus in PDF only on Docsity!

  1. Consider the region S in the plane of all points which are be- tween the graphs of the parabola y = 9 − x^2 and the straight line y = x + 3. The region S is shown in the figure to the right.

1A. Set up the integral which represents the area of S if the corre- sponding approximations are rectangles each of whose base width is 4 x and each rectangle goes from a top curve down to a bottom curve. That is, the integral is of the form

dx.

1B. Evaluate the integral in (1A) to find that area. Show all your work.

2A. Set up the integral which gives the arc length of the curved part of the boundary of S, that is, the arc length of the graph of y = 9 − x^2 from x = −3 to x = 2.

2B. The integral in 2A is “doable” — the back of your book has a formula for the antiderivative you’d need. But it’s so complicated, and in practice, a good numerical approximation will do. Indeed, find the MID(50) approximation for the integral in 2A.

  1. Again, consider the region S from problem 1, of all points which are between the graphs of the parabola y = 9 − x^2 and the straight line y = x + 3. The region S is shown in the figure to the right.

3A. Set up the integral(s) which represents the volume of the solid of revolution obtained by revolving S around the line y = −1. (Do not evaluate the integral(s)).

3B. Now set up the integral(s) which represents the volume of the solid of revolution obtained by revolving S around the line x = −3. (Do not evaluate the integral(s)).

  1. The following table of velocities v(t) in feet per second at various times t for some moving object was recorded during an experiment: t 0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5. v(t) -27 -21.1 -15.4 -10.3 -5.8 -2.0 1.3 4.0 6.4 8.4 10.

5A. What is the physical meaning of

1 v(t)^ dt?

5B. For the integral

1 v(t)^ dt^ in (5A), estimate LHS(n), RHS(n), TRAP(n) and MID(n) for the maximum possible number of subintervals n in each case, using only the information available in the table. Clearly label all your answers!

5C. The table suggests that v(t) is an increasing, concave-down function on [0, 5]. Suppose it is, and there was enough

information to find I =

0 v(t)^ dt^ for each of LHS(n), RHS(n), TRAP(n) and MID(n) with^ n^ = 25.^ From smallest to largest, put these numbers in order: I, LHS(n), RHS(n), TRAP(n) and MID(n).

6A. Find

1

cos (

x ) √ x

dx by the method of substitution, using appropriate notation throughout.

6B. In particular, what are the limits on the integral in (6A) after the appropriate substitution is made?