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

If this part uses an earlier result, revisit (c)i Parametric curves or open all working together.

(c)ii

Parametric area

The line LL with equation y=−2x−5y=-2x-5 meets CC at (−2,−1)(-2,-1). The region SS is enclosed by LL, CC and the xx-axis. Find the exact area of SS.

Explore this idea

Hint

  1. SS lies below the xx-axis. Is the signed integral going to be negative?

  2. Area = (area between CC and the axis) minus a small triangle cut off by LL.

  3. Convert ∫y dx\int y\,dx to tt and simplify with sin⁡2+cos⁡2=1\sin^2+\cos^2=1 and 2sin⁡cos⁡=sin⁡2\sin\cos=\sin.

Answer · (c)ii

A=π/2−5/4A=\pi/2-5/4.

Explore the idea

Open interactive exploration

A sliver below the axis, cut off by the line

C is an arc of the ellipse x²/8 + y²/2 = 1. Between x = −2√2 and x = −5/2 the top of S is the axis; from −5/2 to −2 the top is the line L.

Number of stripsn = 4
Try an example

Where does one strip’s area come from?

1 of 4

The curves meet at this sample, so the strip has zero height and contributes zero. Its red marks use the left edge at x* = -2.8284.

zero heightupper = lowerwidth Δx
Zoomed-in rectangle: width and height are scaled independently. The sample sets its upper and lower values across the whole strip.
Sample x*
-2.8284
Width Δx
0.20711
Upper value
0
Lower value
0
Height: upper − lower
0
Strip area
0

0.20711 ×(0 − 0)≈ 0

Add this contribution to those of the other 3 strips to get the sum below. Displayed values are rounded.

Strip width
0.207
0.828 ÷ 4
Sum of strips
0.28893
Exact, π/2 − 5/4
0.32080
Error
−0.03187
9.94% too small

How the sum improves as n grows

  • C, part of the ellipse
  • Top of S: the x-axis, then the line L from (−5/2, 0) down to (−2, −1)
  • Height = top − bottom
  • Exact area π/2 − 5/4 ≈ 0.3208
Key idea

The parametric integral reduces to ∫(1+sin⁡t) dt\int(1+\sin t)\,dt, which gives π2−1\frac\pi2-1 for the area under the arc. Subtracting the triangle gives π2−54≈0.321\frac\pi2-\frac54\approx0.321.

Ideas to try
  1. Find the kink at x = −5/2 where the top switches from the axis to the line.
  2. Compare midpoints with n = 10.