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Section 4.8

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1. (a) Distance \displaystyle{= \int_0^3 |32t|dt = 144}

(c) Distance \displaystyle{= \int_0^2 |t^2 - t - 6|dt = \frac{34}{3}}

(e) Distance \displaystyle{= \int_0^\pi |2\sin(2t)|dt = 4}

3. (a) \displaystyle{x(t) = \int_0^t (3s^2 - 6)ds + 5 = t^3 - 6t + 5}

4. \displaystyle{x(t) = -\int_0^t x_0\sqrt{\frac{k}{m}}\sin\left(\sqrt{\frac{k}{m}}s\right)ds + x_0 = x_0\cos\left(\sqrt{\frac{k}{m}} \ t\right)}

6. (a) \displaystyle{x(t) = \int_2^t (3s^3 + 6s - 17)ds = \frac{3}{4}t^4 + 3t^2 - 17t + 14}

(c) \displaystyle{x(t) = \int_0^t \sin^2(2s)ds = \frac{1}{2}t - \frac{1}{8}\sin(4t) + 2}

(e) \displaystyle{x(t) = \int_4^t \sqrt{1 + 2s} \ ds + 3 = \frac{1}{3}(1 + 2t)^\frac{3}{2} - 6}

8. (a) \displaystyle{x(t) = -16t^2 + 20t + 100} feet

(c) Maximum height \displaystyle{= x\left(\frac{5}{8}\right) = 106.25} feet

9. (a) Length \displaystyle{= \int_0^2 \sqrt{1 + 9x} \ dx = \frac{2}{27}(19\sqrt{19} - 1)}

(c) Length \displaystyle{= \int_{-1}^1 \sqrt{1 + 9x^4} \ dx = 3.09573}, rounded to five decimal places

(e) Length \displaystyle{= \int_0^\pi \sqrt{1 + \sin^2(2t)} \ dt = 3.82020}, rounded to five decimal places

10. The flat sheet would need to be 14.64 feet long.

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