[These are confirmations rather than proofs, because the calculus of trigonometric functions was developed on the basis of the formulae in parts a and b.] a Use integration to confirm that the area of a circle is ar?. (Hint: Find the area bounded by the semi-circle y = Vr2 x? and the x-axis and double it. Use the substitution x = = r sin 0.) b The shaded area in the diagram to the right is the segment of a circle of radius r cut off by the chord AB subtending an angle a at the centre O. A Vr2 - x² dx. r cos a i Show that the area is I = 2 "COs ii Let x = r cos 0, and show that I -2r2 sin? 0 do. %3D B. iii Hence confirm that I = }r? (a – sin a).
[These are confirmations rather than proofs, because the calculus of trigonometric functions was developed on the basis of the formulae in parts a and b.] a Use integration to confirm that the area of a circle is ar?. (Hint: Find the area bounded by the semi-circle y = Vr2 x? and the x-axis and double it. Use the substitution x = = r sin 0.) b The shaded area in the diagram to the right is the segment of a circle of radius r cut off by the chord AB subtending an angle a at the centre O. A Vr2 - x² dx. r cos a i Show that the area is I = 2 "COs ii Let x = r cos 0, and show that I -2r2 sin? 0 do. %3D B. iii Hence confirm that I = }r? (a – sin a).
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
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Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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![12 [These are confirmations rather than proofs, because the calculus of trigonometric functions was
developed on the basis of the formulae in parts a and b.]
a Use integration to confirm that the area of a circle is ar?.
Vr2
- x² and the x-axis and double it. Use the
(Hint: Find the area bounded by the semi-circle y
r sin 0.)
-
substitution x =
b The shaded area in the diagram to the right is the segment of a circle of radius
r cut off by the chord AB subtending an angle a at the centre O.
A
L Vr? - x² dx.
i
Show that the area is I = 2
r cos
ii Let x = r cos 0, and show that I = -2r²
sin? 0 d0.
1
В
iii Hence confirm that I =
r2 (α-sin α).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F70beab56-a43b-4c48-9bfa-d1f4d172e84f%2F8da1da96-73f4-49a8-8d7c-0ed9e0a3e9bf%2Fq9sojwe_processed.jpeg&w=3840&q=75)
Transcribed Image Text:12 [These are confirmations rather than proofs, because the calculus of trigonometric functions was
developed on the basis of the formulae in parts a and b.]
a Use integration to confirm that the area of a circle is ar?.
Vr2
- x² and the x-axis and double it. Use the
(Hint: Find the area bounded by the semi-circle y
r sin 0.)
-
substitution x =
b The shaded area in the diagram to the right is the segment of a circle of radius
r cut off by the chord AB subtending an angle a at the centre O.
A
L Vr? - x² dx.
i
Show that the area is I = 2
r cos
ii Let x = r cos 0, and show that I = -2r²
sin? 0 d0.
1
В
iii Hence confirm that I =
r2 (α-sin α).
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