Section 4.1 « Difference Equations to Differential Equations

Section 4.1

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1. (a) The area is between $\dfrac{77}{60}$ and $\dfrac{25}{12}$ .

(b) The area is between $14$ and $30$ .

(c) The area is between $6$ and $14$ .

(d) The area is between $\dfrac{\sqrt{2}\pi}{4}$ and $\dfrac{(2 + \sqrt{2})\pi}{4}$ .

2. (a) $\displaystyle{\frac{81}{4} \le \int_1^4 3xdx \le \frac{99}{4}}$

(c) $\displaystyle{-\frac{2\pi}{3} \le \int_{-\pi}^\pi \cos(x)dx \le \frac{2\pi}{3}}$

(e) $\displaystyle{-0.3072 \le \int_0^1 (x^3 - x)dx \le -0.1813}$

3. (a) $A_R = 30$ , $A_L = 14$

(c) $A_R = \dfrac{496}{315}$ , $A_L = \dfrac{352}{105}$

(d) $A_R = \dfrac{25}{64}$ , $A_L = \dfrac{9}{64}$

4. (a) $A_R = 0.33835$ , $A_L = 0.32835$

(c) $A_R = 4.08040$ , $A_L = 3.92040$

(e) $A_R = 0.69065$ , $A_L = 0.69565$

(g) $A_R = 1.76281$ , $A_L = 1.76281$

5. (a) $A_R = \dfrac{15}{4}$ , $A_L = \dfrac{7}{4}$

(c) $A_R = 0$ , $A_L = 0$

(e) $A_R = -\dfrac{5}{4}$ , $A_L = -\dfrac{5}{4}$

6. (a) $A_R = 9.13545$ , $A_L = 8.86545$

(c) $A_R = 6.27654$ , $A_L = 6.27654$

(e) $A_R = -1.33320$ , $A_L = -1.33320$

(g) $A_R = -0.34407$ , $A_L = -0.34907$

7. (a) $\displaystyle{\int_0^4 xdx = 8}$

(c) $\displaystyle{\int_0^3 \sqrt{9 - x^2} \ dx = \frac{9\pi}{4}}$

(d) $\displaystyle{\int_{-2}^2 4\sqrt{4-t^2} \ dt = 8\pi}$

(e) $\displaystyle{\int_{-2}^2 x^3dx = 0}$

8. (a) $\displaystyle{\int_0^1 f(x)dx = \frac{3}{2}}$

(c) $\displaystyle{\int_0^3 f(x)dx = \frac{19}{2}}$

12. (c) $F'(x) = f(x)$

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