[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).

Q: ㅠ Using the trigonometric substitution, = sin 0, given 0 ≤ 0 < 2 trigonometric function: x Simplify…

A:

Q: Use the mean value theorem to prove the inequality for all a and b: Isina – sinb| < |a - b|.

A:

Q: Find all values of 0 in the interval [0°,360°) that have the given function value. tan 0 = /3 (Type…

A: given

Q: Determine whether the following statementsare true and give an explanation or counter example.Graph…

A: Given statement ∫02πasinaxdx=∫02πacosaxdx=0

Q: IfØ is the angle between a line through the origin and the positive x-axis, the area, in cm?, of…

A: Given that: A=3θ+4sinθ+12sin2θ Therefore, rate of change of area charging is as following: dAdt.…

Q: Integrate the function, by making an appropriate trigonometric substituti 1 dx √81-x² 2) Ś

A: Given: ∫01dx81-x2

Q: 3. Differentiate the following and simplify. a) y = x²-1 x²+1 b) y = sec x tan x

A: (a)

Q: Use a trigonometric substitution to calculate a non-zero constant. X √x² + 2ax dx, where a is

A: The given integration is: ∫xx2+2axdx Use a trigonometric substitution to find the value of the given…