Question 4 (a) Calculate the following: i. the area of the region in the first quadrant bounded by the curves of f(x)=√x and g(x) = x − 2, and the x-axis. Show your working. ii. the area of the region bounded by the curves of f(x) = sin r and g(x) = cosa in the interval []. Show your workin I

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Chapter2: Second-order Linear Odes
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Needed to be solved Part A Correctly in 15 minutes and get the thumbs up please show neat and clean work for it
Question 4
(a) Calculate the following:
i. the area of the region in the first quadrant bounded by the curves of
f(x)=√x and g(x)=x-2, and the x-axis. Show your working.
ii. the area of the region bounded by the curves of f(x) = sin and
g(x) = cosa in the interval [4, 5]. Show your workin
(b) Using the Taylor expansion of arctan x,
1<x< 1, calculate the sum
1
3n (2n +1)
Σ(-1)",
n=0
I
(c) Let f: [0, +∞o) → R be continuously differentiable. Show that there exist
continuous, positive and increasing functions g, h: [0, +∞) → R such
that f=g-h.
Hint: You may use the fact that for continuous functions g, h: [0, +∞) →
R, it holds that max{g, h}
g+h+g-hand min{g, h}
g+h-lg-hare
also continuous.
Transcribed Image Text:Question 4 (a) Calculate the following: i. the area of the region in the first quadrant bounded by the curves of f(x)=√x and g(x)=x-2, and the x-axis. Show your working. ii. the area of the region bounded by the curves of f(x) = sin and g(x) = cosa in the interval [4, 5]. Show your workin (b) Using the Taylor expansion of arctan x, 1<x< 1, calculate the sum 1 3n (2n +1) Σ(-1)", n=0 I (c) Let f: [0, +∞o) → R be continuously differentiable. Show that there exist continuous, positive and increasing functions g, h: [0, +∞) → R such that f=g-h. Hint: You may use the fact that for continuous functions g, h: [0, +∞) → R, it holds that max{g, h} g+h+g-hand min{g, h} g+h-lg-hare also continuous.
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