3 free questions from this chapter (the full set has 30). Marking: Multi-correct: +4 all correct, +1 per correct option if none wrong, −2 if any wrong · Numerical: +4 / 0 · Matching: +3 / −1.
Question 1Level 2 · ExamSingle correct
Let C be the curve y=x3−x and let T be the tangent to C at the point (−1,0). The area of the bounded region enclosed by C and T is ______. Exactly one option is correct.
A227
B427
C49
D4
Show answer and solution
Answer: B
y′=3x2−1 equals 2 at x=−1, so T:y=2x+2.
Intersections: x3−x=2x+2⟺x3−3x−2=0⟺(x+1)2(x−2)=0, so x=−1 (the point of tangency, a double root) and x=2 (the point (2,6)).
Sign analysis: (2x+2)−(x3−x)=−(x+1)2(x−2)≥0 on [−1,2], so T lies above C there and the bounded region is −1≤x≤2, x3−x≤y≤2x+2.
Area =∫−12(−x3+3x+2)dx=[−4x4+23x2+2x]−12=6−(−43)=427.
Insight: for a monic cubic and its tangent at α meeting it again at β, the difference is ±(x−α)2(x−β) and the area is 12(β−α)4=1281=427.
Trap: 227 is twice the area.
Difficulty: Hard
Question 2Level 3 · AdvancedSingle correct
The area of the region bounded by the curves y=sinx, y=cosx and the lines x=0, x=23π is ______. Exactly one option is correct.
A2
B42
C22
D42−2
Show answer and solution
Answer: D
Crossings in [0,23π]: sinx=cosx⟺tanx=1, i.e. x=4π,45π.
Sign analysis of cosx−sinx=2cos(x+4π): positive on [0,4π), negative on (4π,45π), positive on (45π,23π].
With F(x)=sinx+cosx (an antiderivative of cosx−sinx): F(0)=1, F(4π)=2, F(45π)=−2, F(23π)=−1.
Pieces: 2−1, then 22, then −1+2. Area =42−2.
Trap: the signed integral ∫03π/2(sinx−cosx)dx=2 is not the area; the pieces must be taken with absolute values.
Difficulty: Very hard
Question 3Level 3 · AdvancedSingle correct
Let R={(x,y):x2+y2≤4,x2≤3y}. The area of R is ______. Exactly one option is correct.
A34π+33
B34π−33
C38π−33
D32π+3
Show answer and solution
Answer: A
Boundary curves: circle x2+y2=4 and parabola y=3x2. Intersection: 3y+y2=4⇒y=1 (the root y=−4 is impossible since y=x2/3≥0), so x=±3.
Sketch: R is the cap of the disc above the parabola, symmetric about the y-axis; for ∣x∣≤3 the upper boundary is y=4−x2 and the lower is y=3x2 (at x=0: 2>0).
Area =2∫03(4−x2−3x2)dx=2[2x4−x2+2sin−12x]03−92(3)3=2(23+32π)−323=34π+33.
Trap: 38π−33 is the rest of the disc (4π minus the answer).
Difficulty: Very hard
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