Use identities for the integral and geometry for the regions

Graphs y=e²ˣ and y=(x+1)², intersection (−1.2785,0.07754), common y-intercept 1Source-given figure redrawn with D3. Graphs y=e²ˣ and y=(x+1)², intersection (−1.2785,0.07754), common y-intercept 1. Only labels supplied in the original question are included.xyO(−1.2785, 0.07754)1y=(x+1)²y=e²ˣ
Question diagram 1
Graphs y=e²ˣ and y=(x+1)², intersection (−1.2785,0.07754), common y-intercept 1Source-given figure redrawn with D3. Graphs y=e²ˣ and y=(x+1)², intersection (−1.2785,0.07754), common y-intercept 1. Only labels supplied in the original question are included.xyO(−1.2785, 0.07754)1y=(x+1)²y=e²ˣ

Given exponential and quadratic graphs with printed intersection and common intercept.

Curve C and line L enclosing shaded S; labelled intersection (−2,−1)Source-given figure redrawn with D3. Curve C and line L enclosing shaded S; labelled intersection (−2,−1). Only labels supplied in the original question are included.xyOCSL(−2, −1)
Question diagram 2
Curve C and line L enclosing shaded S; labelled intersection (−2,−1)Source-given figure redrawn with D3. Curve C and line L enclosing shaded S; labelled intersection (−2,−1). Only labels supplied in the original question are included.xyOCSL(−2, −1)

Given curve C and line L enclosing shaded region S, with printed intersection.

(a) Find ∫(cot⁡62x+cot⁡42x−sin⁡(x/2)sin⁡(3x/2))dx\int(\cot^62x+\cot^42x-\sin(x/2)\sin(3x/2))dx. marks

(b) The diagram shows y=e2xy=e^{2x} and y=(x+1)2y=(x+1)^2, intersection (−1.2785,0.07754)(-1.2785,0.07754) and yy-intercept 1. R is the finite region bounded by both curves. Find the volume when R is rotated 2π2\pi about the yy-axis, to 4 decimal places. marks

(c) C: x=2(sin⁡(t/2)−cos⁡(t/2))x=2(\sin(t/2)-\cos(t/2)), y=sin⁡(t/2)+cos⁡(t/2)y=\sin(t/2)+\cos(t/2), −3π/2≤t≤−π/2-3\pi/2\le t\le-\pi/2. Line L: y=−2x−5y=-2x-5, meets C at (−2,−1)(-2,-1). S is enclosed by L,C and the xx-axis as shown.

(i) Show C’s xx-intercept is −22-2\sqrt2. marks

(ii) Find the exact area of S. marks

[2020/ASRJC/Prelim/II/4]

Total marks: 14

Solution

(a)

Trigonometric integration

Find ∫(cot⁡62x+cot⁡42x−sin⁡x2sin⁡3x2)dx\displaystyle\int\left(\cot^6 2x+\cot^4 2x-\sin\tfrac x2\sin\tfrac{3x}2\right)dx.

Hint

  1. Group cot⁡6+cot⁡4\cot^6+\cot^4 and factor. Is there an identity that links cot⁡2\cot^2 and csc⁡2\csc^2?

  2. Turn the product sin⁡sin⁡\sin\sin into a difference of cosines.

  3. Integrate each part: one is a reverse chain rule.

Answer · (a)

−cot⁡5(2x)/10+sin⁡2x/4−sin⁡x/2+C-\cot^5(2x)/10+\sin2x/4-\sin x/2+C.

On intervals with sin2x≠0.