Chapter 6, Answers

Chapter 6

6.1.5

1. (a) \( R\left(\pi f^2,n\right)=\frac{3\pi }{n}\left(n +\frac{3}{n}\frac{n(n+1)}{2}\right)\);

   (b) \(\int_{-1}^2 \pi \left( \sqrt{x}\right)^2\, dx =
   \lim_{n\to \infty}\sum _{i=1}^n\pi\left(\sqrt{x_i}\right)^2\Delta x
   = \frac{15}{2}\pi\).

2. (a) \( R\left(\pi f^2,n\right)=\frac{27\pi }{n}\left(\frac{4}{n^2}\frac{n(n+1)(2n+1)}{6}+\frac{4}{n}\frac{n(n+1)}{2}+n\right)\);

   (b) \(\int_{1}^4 \pi \left( 2x+1\right)^2\, dx =
   \lim_{n\to \infty}\sum _{i=1}^n\pi\left( 2x_i+1\right)^2\Delta x
   = 117\pi\).

3. \(V(\mathcal{S})=\frac{86 \pi }{3}\).
4. (a) \( R\left(\pi f^2,n\right)=\pi \left(\frac{2}{n}\right)^5\left(\frac{1}{30} n (n+1) (2 n+1) \left(3
   n^2+3 n-1\right)\right)\);

   (b) \(\int_{0}^2 \pi \left( x^2\right)^2\, dx =
   \lim_{n\to \infty}\sum _{i=1}^n\pi\left( x_i^2\right)^2\Delta x
   = \frac{32 \pi }{5}\).

5. (a) \( R\left(\pi f^2,n\right)=\pi \sum _{i=1}^n (f\left(x_i\right))^2\Delta x=\pi \sum_{i=1}^n \sin ^2\left(\frac{\pi i}{n}\right)\frac{\pi }{n}\); (b) \(\frac{\pi^2}{2}\).
6. (a) \( R\left(\pi f^2,n\right)=\pi \sum _{i=1}^n (f\left(x_i\right))^2\Delta x=\pi \sum_{i=1}^n \left (\cos \left(-\pi+\frac{\pi i}{n}\right)+2\right )^2\frac{\pi }{n}\); (b) \(6\pi^2\).
7. (a) \( R\left(\pi f^2,n\right)
   =\pi \left(\frac{2}{n}\right)^7\left(\frac{1}{42} n (n+1) (2 n+1)
   \left(3n^4 + 6n^3- 3 n+1\right)\right)\);

   (b) \(\frac{64 \pi}{21}\left(1+\frac{1}{n}\right)\left(2+\frac{1}{n}\right)
   \left(3 + 6 \frac{1}{n}-\frac{3}{n^3}+\frac{1}{n^4}\right)\underset{n\to
   \infty }{\to }\frac{128 \pi }{7}\).

8. \(\frac{3544 \pi }{105}\).

6.2.6

1. (a) \(32/3 \); (b) \(8\sqrt{3}/3 \); (c) \(4\pi/3 \).
2. (a) \( 16/3\); (b) \(4\sqrt{3}/3 \); (c) \(2\pi/3 \).
3. (a) \( 128/3\); (b) \(16 \pi/3 \); (c) \( 64/3\).
4. (a) \( 16/3\); (b) \(8\sqrt{3} \); (c) \(4 \pi\).
5. (a) \( 4/3\); (b) \( 2\sqrt{3}\); (c) \(\pi/6 \).
6. (a) \(1/30 \); (b) \( \sqrt{3}/20\); (c) \(\pi/240 \).

For problems 7—10,
\(Vol(\mathcal{S})= \int_{a}^{b}A(y)\,dy = \lim_{n\to \infty} \sum _{i=1}^n A\left(y_i\right)\,\Delta y\).

7. \(y\in [a,b]=[0,\pi]\), \(\Delta y =\frac{\pi}{n}\), \(y_i=\frac{\pi i}{n}\).
   (a) \(A(y)=\sin^2 y\);
   (b) \(A(y)=\frac{\sqrt{3}}{4}\sin^2 y\);

   (c) \(A(y)=\frac{\pi}{8}\sin^2 y\).

8. \(y\in [a,b]=[0,1]\), \(\Delta y =\frac{1}{n}\), \(y_i=\frac{i}{n}\).
   (a) \(A(y)=\frac{1}{4}\left(\sqrt{y}-y^2\right)^2\);
   (b) \(A(y)=\frac{\sqrt{3}}{4}\left(\sqrt{y}-y^2\right)^2\);

   (c) \(A(y)=\frac{\pi}{8}\left(\sqrt{y}-y^2\right)^2\).

9. \(y\in [a,b]=[0,\pi/4]\), \(\Delta y =\frac{\pi}{4n}\), \(y_i=\frac{\pi i}{4n}\).
   (a) \(A(y)=\tan^2 y\);
   (b) \(A(y)=\frac{\sqrt{3}}{4}\tan^2 y\);

   (c) \(A(y)=\frac{\pi}{8}\tan^2 y\).

10. \(y\in [a,b]=[0,1]\), \(\Delta y =\frac{1}{n}\), \(y_i=\frac{i}{n}\).
    (a) \(A(y)=\left(\frac{\pi}{4}-\arctan y\right)^2\);

    (b) \(A(y)=\frac{\sqrt{3}}{4}\left(\frac{\pi}{4}-\arctan y\right)^2\);
    (c) \(A(y)=\frac{\pi}{8}\left(\frac{\pi}{4}-\arctan y\right)^2\).

6.3.5

1. (b) \(\frac{28\pi}{3}\).
2. (b) (i) \(\frac{\pi}{2}\); (ii) \(\frac{5\pi}{6}\).
3. (b) (i) \(\frac{24\pi}{5}\); (ii) \(\frac{248\pi}{15}\).
4. (b) (i) \(\frac{\pi}{6}\); (ii) \(\frac{\pi}{2}\).
5. \(9\pi^2\).
6. (b) (i) \(\frac{8\pi}{3}\); (ii) \(\frac{16\pi}{15}\).
7. \(\pi\left(4 + \frac{3\pi}{2}\right )\).

6.4.3

1. (b) (i) \( \frac{3\pi }{10}\); (ii) \( \frac{31\pi }{30}\).
2. (b) (i) \( \frac{1 }{5}\left(240 +79\pi\right)\); (ii) \( \frac{1 }{5}\left(80 +27\pi\right)\).
3. (b)
   (i) \( \frac{\pi }{6\ln 2}\left(40\ln 2- 27\right)\);
   (ii) \( \frac{\pi }{6\ln 2}\left(32\ln 2- 21\right)\).
4. (b)
   (i) \( \frac{1 }{24}\left(48-13\pi\right)\);
   (ii) \( \frac{1 }{24}\left(-48+\pi + 96\ln 2\right)\);
5. (b) (i) \(\frac{63\pi}{2}\); (ii) \(\frac{108\pi}{5}\); (iii) \(\frac{117\pi}{5}\); (iv) \(\frac{27\pi}{2}\).
6. \(\frac{2}{3}\left (4\pi-3\sqrt 3\right )\).
7. (a) \(\frac{29\pi}{30}\); (b) \(\frac{11\pi}{30}\).
8. (a) \(\frac{3\pi}{4}\); (b) \(\frac{5\pi}{4}\).
9. (a) \(\frac{1}{4}\pi(2e^{-1/2} + 16 e^{-1/4} – 27 + 8e^{1/2} + e)\); (b) \(\frac{1}{4}\pi(-2e^{-1/2} + 16 e^{-1/4} – 21 + 8e^{1/2} – e)\).
10. (a) \(\frac{1}{6}\pi\left (6\sqrt{3}+\pi\right )\);
    (b) \(\frac{1}{6}\pi\left (7\pi – 6\sqrt{3}\right )\).

6.5.2

1. (a) Scanning axis, a vertical \(z\)-axis. \( A(z) = \frac{ab}{2c^2}(c-z)^2\), \(z \in [0,c]\);
   (b) \(\int_{0}^{c}A(z)\,dz
   =\lim_{n\to \infty} \sum _{i=1}^n A\left(z_i\right)\,\Delta z \),
   with \(\Delta z = \frac{c}{n}\), \(z_i= \frac{ci}{n}\);
   (c) \(\frac{abc}{6}\).
2. (a) Scanning axis, a vertical \(z\)-axis. \(A(z) = \frac{(b-a)^2}{h^2}(h-z)^2\), \(z \in [0,h]\);

   (b) \(\int_{0}^{h}A(z)\,dz
   =\lim_{n\to \infty} \sum _{i=1}^n A\left(z_i\right)\,\Delta z \),
   with \(\Delta z = \frac{h}{n}\), \(z_i= \frac{h i}{n}\);
   (c) \( \frac{a^2+ab+b^2}{3h}\).

3. Scanning axis, the \(y\)-axis. (a)\(A(y)=4\pi R\sqrt{r^2-y^2}\), \(y \in [-r,r]\);

   (b) \( \int_{-r}^{r}A(y)\,dy = \lim_{n\to \infty} \sum _{i=1}^n A\left(y_i\right)\,\Delta y\),
   with \(\Delta y =\frac{2r}{n}\), \(y_i=\frac{2r i}{n}\).
   (c) \((2\pi R)(\pi r^2)\).

4. (a) Scanning axis, a vertical \(z\)-axis. \( A(z) = 2l \sqrt{2^2-z^2}\),
   with \(z \in [-2,1]\);

   (b) \(\int_{-2}^{1}A(z)\,dz
   =\lim_{n\to \infty} \sum _{i=1}^n A\left(z_i\right)\,\Delta z \),
   with \(\Delta z = \frac{3}{n}\), \(z_i= -r+\frac{3 i}{n}\);
   (c) \(\left(\sqrt{3}+\frac{8\pi}{3}\right)l\).

5. Make sure you understand Example 1 in 6.5.1 and problem #4.