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Our professor gave us these two problems and 24 hours to do it. But I'm struggling with finding the answers.

Extra credit

First question is about finding the volume of a shape with respect to y=$\frac{\sqrt{3}}{2}$ I don't know which method to use shell or washer?

And I know that the second question can be solved by taking the derivative and finding the limits when x approaches infinity which would be 1/e but I'm still don't get the logic behind it.

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    For the first question, is it volume or area? Is there some rotation involved? And if yes, about which axis? – 2017-02-08
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    Yes it is volume. Rotation is with respect to y= $\frac{\sqrt{3}}{2}$ – 2017-02-08
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    See the edit on my answer. – 2017-02-08

2 Answers 2

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Hint for first question: Draw lines from the origin to the points $\left(\pm \frac{1}{2}, \frac{\sqrt{3}}{2} \right)$. This shows the shaded area as a circular sector with angle $\pi/3$ minus a triangle. In 3D, this means that the volume you want is a spherical sector minus a cone.

Hint for second question: either L'Hopital, or expand the expression with the binomial theorem as $\sum_{n=0}^x x^{-n} {x \choose n}$, and find the limit of each term as $x \to \infty$. This should give you a familiar power series.

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    for the second question how do I use L'Hopital rule? – 2017-02-08
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    Use $\lim_{x \to \infty} \ln f(x) = \ln \left( \lim_{x \to \infty} f(x) \right)$, and note that $\ln f(x) = x \ln \left( 1 + \frac{1}{x} \right)$ is the product of a term that goes to infinity and a term that goes to zero. – 2017-02-08
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    Your hint for the first question really helped me thanks a lot! I calculated the volume of the cone and it came out to be $\frac{^{\pi }}{12}$ if I'm not mistaken everything is perfect so far. The only part I don't get it is how do I find the spherical sector and what does 3D mean. Sorry I'm an international student please excuse my english. – 2017-02-08
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    I solved the second question with your hint thanks for that too :) – 2017-02-08
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    Try this article: https://en.wikipedia.org/wiki/Spherical_sector – 2017-02-09
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Hint: For $(1)$, any of the methods can work if used correctly. That said, I suggest integrating disks (think 'coins'), parallel to the $y$-axis and centered on $y=\frac{\sqrt{3}}2$.

For $(2)$, prove that if $f$ is positive then

$$f \text{ is increasing}\iff \ln(f) \text{ is increasing}$$

Can you solve the limit using L'Hopital?

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    So if I were to use the washer method would the integral be $\pi \int _a^b\:\left(outer\:radius\right)^2-\left(inner\:radius\right)^2dx$ but how do I plug in the values that I have Please help me out I'll miss the chance to get an a if i fail to do this problem.. – 2017-02-08
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    I don't know what the washer method is. Moreover, I *am* helping you out, but I won't do your homework. – 2017-02-08
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    I did not ask you to do my homework. – 2017-02-08