Applications of Integration — Book-Back Answer Key

Applications of Integration — Book-Back Answer Key (Exercise 9.10)

Each entry gives the correct option and a one-line reason. Option letters map as A = 1, B = 2, C = 3, D = 4.

1. (A) \(\tfrac{\pi}{6}\): pull out the 9 to reach \(\tfrac13\sin^{-1}\) form; value \(\tfrac13\cdot\tfrac{\pi}{2}\).
2. (C) \(\tfrac{5}{2}\): split \(|x|\) at \(0\); areas \(\tfrac12+2\).
3. (C) \(0\): \(x\mapsto\pi-x\) flips the odd-multiple cosine cube, so \(I=-I\).
4. (D) \(\tfrac{2}{3}\): integrand is even; \(2\int_0^1 u^2\,du\) with \(u=\sin x\).
5. (D) \(4\pi\): \(\tan^{-1}t+\tan^{-1}\tfrac1t=\tfrac{\pi}{2}\), constant over length \(8\).
6. (C) \(2\): odd \(\times\sec^2 x\) terms vanish; only \(\int\sec^2 x\) survives.
7. (C) \(x\cos x\): Fundamental Theorem — derivative of the integral is the integrand.
8. (C) \(\tfrac{8}{3}\): \(a=1\), latus rectum \(x=1\); \(2\int_0^1 2\sqrt{x}\,dx\).
9. (B) \(\tfrac{1}{10100}\): beta formula \(\tfrac{1!\,99!}{101!}=\tfrac{1}{101\cdot100}\).
10. (A) \(\tfrac{\pi}{2}\): add the \(x\mapsto\pi-x\) copy; \(2I=\pi\).
11. (D) \(9\): \(\tfrac{\Gamma(n+2)}{\Gamma(n)}=n(n+1)=90\Rightarrow n=9\).
12. (B) \(\tfrac{2}{9}\): \(u=3x\) then Wallis \(\int_0^{\pi/2}\cos^3=\tfrac23\), times \(\tfrac13\).
13. (B) \(\tfrac{3\pi}{8}\): symmetry doubles \(\int_0^{\pi/2}\sin^4=\tfrac{3\pi}{16}\).
14. (D) \(\tfrac{2}{27}\): gamma integral \(\tfrac{2!}{3^3}\).
15. (D) \(2\): \(\tfrac12\tan^{-1}\tfrac{a}{2}=\tfrac{\pi}{8}\Rightarrow a=2\).
16. (D) \(\tfrac{\pi a^3}{6}\): \(V=\pi\int_0^a (ax-x^2)\,dx\).
17. (C) \(9\): \(t=x^2\) sends the integral to \(\tfrac12[f(9)-f(1)]\).
18. (D) \(\tfrac{\pi^2}{4}-2\): \(x=\sin\theta\) then by-parts twice on \(\theta^2\cos\theta\).
19. (B) \(\tfrac{3\pi a^4}{16}\): \(x=a\sin\theta\) gives \(a^4\int_0^{\pi/2}\cos^4\theta\).
20. (A) \(\tfrac{1}{2}\): differentiate to get \(f(x)=\tfrac{1}{1+x}\), so \(f(1)=\tfrac12\).