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I was wondering if someone could take a look at two equations to see if I am solving them properly. I'm currently in Calculus II, but is been a year since I took Calculus I (budget cuts cancelled the Calc II class last spring,) so I want to make sure I still haven't forgotten anything.

These questions came from my Calc II professor's final exam she gave last year to her Calc I class. She gave us the exam for homework to see where our skills were. I believe these are correct, but I wanted to make sure my work was correct. Anyway, here they are (my apologies for a long post):

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Question 1: Consider the region bounded by the graph [tex] y = 2x - x^2[/tex] and [tex] y = x [/tex]

Part 1: Set up the integral for the volume of the solid formed by revolving the region around the y-axis

Part 2: Find the volume of the solid.

[tex] V = \int_{a}^{b} 2 \pi x f(x)dx [/tex]

[tex] 2x - x^2 = x [/tex]

[tex] x - x^2 = 0 [/tex]

[tex] x (1 - x) = 0 [/tex]

[tex] x = 0 [/tex] and [tex] x = 1 [/tex]

[tex] V = \int_{0}^{1} 2 \pi x (x) dx [/tex]

[tex] V = 2 \pi \int_{0}^{1} x (x) dx [/tex]

[tex] V = 2 \pi \int_{0}^{1} x^2 dx [/tex]

[tex] V = 2 \pi \int_{0}^{1} \frac{1}{3}x^3 dx [/tex]

[tex] V = 2 \pi \left[ \frac{1}{3}x^3 \right]_{0}^{1} [/tex]

[tex] V = 2 \pi \left[ \frac{1}{3}(1)^3 - \frac{1}{3}(0)^3 \right] [/tex]

[tex] V = \frac{2 \pi}{3} [/tex]

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Question 2: Evaluate [tex] \int \frac{\cos \left[ \frac{1}{t} \right]}{t^2} dt [/tex]

[tex] \int \cos \left[ \frac{1}{t} \right] t^{-2} dt [/tex]

let [tex] u = \frac{1}{t} [/tex]

[tex] du = - t^{-2} dx [/tex]

[tex] - \int \cos (u) du[/tex]

[tex] = - sin (u) + C [/tex]

[tex] = - sin \left[ \frac{1}{t} \right] + C [/tex]

Again, my apologies for such a long post.

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# Am I doing this correctly?

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