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Disk and Ring Methods Let f be a continuous function such that f (x) 0 for a x b. Consider the region R under the graph of y = f (x), above the x-axis, and between x = a and x = b (see Fig. 33-1). If R is revolved about the x-axis, the resulting solid is called a solid of revolution. The generating regions R for some familiar solids of revolution are shown in Fig. 33-2.

Capital Account 1,600

Hence, by (2) of Problem 5.15, ICQ2 1 VEE VBEQ1 0:002 T2 25 2 T2 25 =10 ICBO1 RE 1

(b) At T2 1258C, (1) gives us ICQ2 51 5 0:7 0:002 125 25 2 125 25 =10 0:5 10 6 1:53 0:512 2:042 mA 50 3 103

Fig. 33-1 Theorem 33.1: The volume V of the solid of revolution obtained by revolving the region of Fig. 33-1 about the x-axis is given by V =

For the Darlington-pair emitter-follower of Fig. 5-10, nd ICQ1 as a function of the six temperature-sensitive variables ICBO1 ; ICBO2 ; VBEQ1 ; VBEQ2 ; 1 ; and 2 .

Fig. 5-10 By KVL, VCC IBQ1 RF VBEQ1 VBEQ2 IEQ2 RE By KCL, IEQ2 IEQ1 ICQ2 1 (2)

A curve is said to be concave downward if it has the shape of a cap or part of a cap (see Fig. 23-2). In mathematical terms, a curve is concave downward if it lies below the tangent line at an arbitrary point of the curve [see Fig. 23-2(a)].

Using the result of Problem 3.36 in (2) and then substituting IBQ2 IEQ1 , we obtain IEQ2 IEQ1 2 IBQ2 2 1 ICBO2 2 1 IEQ1 2 1 ICBO2 Assuming 1 ; 2 ) 1 and substituting for IEQ1 according to (5.5), we obtain IEQ2 % 2 1 ICQ1 2 1 ICBO2 ICBO1 Also, from (5.6), IBQ1 % ICB1 ICBO1 1 4 3

Close out the drawing account. The drawing account is credited for the total amount of the drawings for the period, and the capital account is deb ited for that amount. The difference between net income and drawing for the period represents the net change in the capital account for the period. The net income of $1,600 less drawings of $400 results in a net increase of $1,200 in the capital account. The closing entry is as follows: Jan. 31 Capital Account 300

Now we substitute (3) and (4) into (1) and rearrange to get ICQ1 VCC VBEQ1 VBEQ2 ICBO1 RF 2 RE ICBO2 2 RE RF = 1 2 RE 5

CHAP. 23]

(a) Determine a rst-order approximation for the change in ICQ1 in the circuit of Fig. 5-10, in terms of the six variables ICBO1 ; ICBO2 ; VBEQ1 ; VBEQ2 ; 1 ; and 2 . (b) Use ICQ1 as found in Problem 5.17 to evaluate the sensitivity factors (that is, the coe cients) in the expression determined in part a.

(a) Since ICQ1 f ICBO1 ; ICBO2 ; VBEQ1 ; VBEQ2 ; 1 ; 2 , its total di erential is given by @ICQ1 @ICQ1 @ICQ1 dI dI dVBEQ1 @ICBO1 CBO1 @ICBO2 CBO2 @VBEQ1 @ICQ1 @ICQ1 @ICQ1 dVBEQ2 d 1 d 2 @VBEQ2 @ 1 @ 2 1

Fig. 23-2 Concavity downward A curve may, of course, be composed of parts of different concavity. The curve in Fig. 23-3 is concave downward from A to B, concave upward from B to C, concave downward from C to D, and concave upward from D to E. A point on the curve at which the concavity changes is called an in ection point. B, C, and D are in ection points in Fig. 23-3.

Using the method of Section 5.3, we may write this as ICQ1 % SI1 ICBO1 SI2 ICBO2 SV1 VBEQ1 SV2 VBEQ2 S 1 1 S 2 2 (b) The sensitivity factors in (1) may be evaluated with the use of (5) of Problem 5.17: @ICQ1 R F 2 R E @ICBO1 RF = 1 2 RE @ICQ1 2 RE @ICBO2 RF = 1 2 RE @ICQ1 @ICQ1 1 SV2 @VBEQ1 RF = 1 2 RE @VBEQ2 @ICQ1 RF VCC VBEQ1 VBEQ2 ICBO1 RF 2 RE ICBO2 2 RE @ 1 RF 1 2 RE 2 @ICQ2 1 RE RF ICBO1 ICBO2 1 VCC VBEQ1 VBEQ2 ICBO1 RF @ 2 RF 1 2 RE 2 2