I (i) The region R is bounded by the x-axis, y = 1 and the curve y= ln (x²) where xeR₁x*0. The area of R may be approximated by the total area, A, of n rectangles each of height as shown in the diagram. 9 1 y 0 1%, 1½ TY₁ 1½ Show that A=2e), where f() is to be determined. 11-0 (II) Find the area of R, giving the answer in exact form. Hence, state y = ln (x²) n limit of A as n→80. (ii) Find the exact volume of the solid formed when R is rotated through 180 about the y-axis.
I (i) The region R is bounded by the x-axis, y = 1 and the curve y= ln (x²) where xeR₁x*0. The area of R may be approximated by the total area, A, of n rectangles each of height as shown in the diagram. 9 1 y 0 1%, 1½ TY₁ 1½ Show that A=2e), where f() is to be determined. 11-0 (II) Find the area of R, giving the answer in exact form. Hence, state y = ln (x²) n limit of A as n→80. (ii) Find the exact volume of the solid formed when R is rotated through 180 about the y-axis.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Transcribed Image Text:(i) The region R is bounded by the x-axis, y = 1 and the curve y=In(x¹) where xe R,x+0. The
1
area of R may be approximated by the total area, A, of n rectangles each of height
in the diagram.
Show that A==
11
-1
y
0
1½
TY/₁
t%
[1½
1
efe
ef(), where f() is to be determined.
y = ln (x²)
n
, as shown
(II) Find the area of R, giving the answer in exact form. Hence, state the limit of A as n→∞o.
(iii) Find the exact volume of the solid formed when R is rotated through 180 about the y-axis.
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