Test Corrections (3/3/14)

All students should correct their tests! You can hand in your corrections, receiving 1/3 of the lost points.

Correct on a separate sheet, NOT on the original. Hand both in, NOT stapled together.

Corrections are to be done on your own – honor code – or with help from me only. No collaboration allowed!

Due Friday (Mar 7) at class time. Extended office hours Thursday from 2:00-5:00.

Clicker Question 1

What is the arc length of y = (4/3) x3/2

from x = 0 to x = 2 ?– A. 52 / 3– B. 13 / 3– C. 9 / 2– D. 18– E. 3

Application: Surface Area

We’ve seen that we can use the integral to compute areas, volumes, and arc lengths by sectioning into small pieces, analyzing their measurements, and then adding up.

We can do the same with surface area. We, again, restrict our study to surface areas

obtained by rotation about a line (usually but not necessarily an axis).

Analyzing a Small Piece

Break the surface into small bands whose width on the axis of rotation is x.

The area of this band will be the distance around times the width of the band, which is not x! What is it in fact?

But we know about that, right?

The Formula and a Classic Example

Hence the surface generated when y = f (x) from a to b is revolved about the x-axis will be the sum of 2(radius of spinning) times the little arc length ds, i.e., it’s

For example, what is the surface area of a sphere of radius r ? Work it out!

b

a

dxdxdyy 2)/(12

Clicker Question 2

An integral representing the surface area generated when the curve y = e x is revolved about the x axis from x = 0 to x = 1 would be

– A. B.

– C. D.

– E. 1

0

212 dxe x

1

0

212 dxex x

1

0

212 dxee xx

1

0

22212 dxex x

1

0

22 1 dxex x

Assignment for Wednesday

Work on test corrections. Learn for your mistakes! Work out the surface area of a sphere. Read Section 8.2. On p. 550, do Exercises 1(a)(i), 3(a)(i), 5, 7. Use Simpson’s Rule with n = 2 to estimate the

surface area generated when y = (1/3)x3 on [0, 4] is revolved around the x-axis.Answer: (2/3)(0 + 42 + 417 + 1284 + 4257) 408