Calculus I, Fall 2014 (http://www.math.nthu.edu.tw/~wangwc/)

Solutions in Quiz05

1. V = ∫_{−1}^{1} π[(2 + \sqrt{1 − y^{2}})^{2} − (2 − \sqrt{1 − y^{2}})^{2}]dy = 8π ∫_{−1}^{1} \sqrt{1 − y^{2}}dy = 8π × \frac{π}{2} = 4π^{2}.

2. S = ∫_{−1}^{1} 2π(2 + \sqrt{1 − y^{2}})\sqrt{1 + \frac{y^{2}}{1−y^{2}}} dy + ∫_{−1}^{1} 2π(2 − \sqrt{1 − y^{2}})\sqrt{1 + \frac{y^{2}}{1−y^{2}}} dy = 8π ∫_{−1}^{1} \frac{1}{\sqrt{1−y^{2}}} dy = 8π^{2}.

3. \frac{dy}{dx} = \sqrt{cos x},

L = ∫0^{\frac{π}{2}} \sqrt{1 + cos x}dx = ∫0^{\frac{π}{2}} \sqrt{2}| cos \frac{x}{2}|dx = 2.

4. e^{y} \frac{dy}{dx} = 3x^{2} ⇒ e^{y} = x^{3} + C for some constant C.

5. \frac{d}{dx}(xy) = x\frac{dy}{dx} + y = e^{x} ⇒ xy = e^{x} + C for some constant C, y(1) = 1 ⇒ y = \frac{1}{x}(e^{x} + 1 − e).