#### MATH 1620

##### Final - Practice 1 | Fall '15
1. Find the area bounded by the line $y = \sqrt{4 - x}$, the $x$-axis and the $y$-axis.

2. Find the area bounded by the $x$-axis and the curve $y = \sin(x)$ on the interval $0 \le x \le \pi$.

3. Use the disc/washer method to find the volume of the solid formed by rotating the regino enclosed by the lines $y = 1, x = 0$ and curve $y = x^3$ around the $x$-axis.

4. Use the shell method to compute the volume of the region formed by rotating the triangle with vertices $(0, 0); (1, 1); (0, 1)$ around the line $x = 1$.

5. Find the volume formed by rotating the region bounded by $y = 2x, x = 1$ and the $x$-axis around the $y$-axis.

6. Find the volume formed by rotating the region bounded by $y = e^x, x = 1$, the $x$-axis and the $y$-axis around the $x$-axis.

7. A $10$ meter chain with mass $100$ kg is suspended vertically from a platform. Use an integral to compute how much work is done lifting the chain onto the platform.

8. A leaky bucket weights $100$ lb when full of water. Suppose water leaks at a rate of $1$ lb per second, and the bucket is lifted at a rate of $2$ ft per second. Write an integral computing the work required to lift the bucket $50$ ft, assuming it is full to start.

9. If $1$ lb of force extends a spring $3$ inches beyond rest length, how much work would be done extending it $6$ inches beyond rest length? Give your answer in foot pounds. Must show appropriate integral and correct answer for full credit.

10. Find the area of the surface formed by rotating the line $y = 2 - x, x = 0$ to $x = 1$ around the $x$-axis.

11. Use integration to find the length of the curve $y = \sqrt{1 - x^2}, x = 0$ to $x = \frac{1}{2}$.

12. Use the Pappus Theorem to find the volume of the solid formed by rotating the diamond-shaped region with corners $(1, 0); (0, 1); (-1, 0); (0, -1)$ around the line $x = 2$.

13. Compute $\int x \sec^2(x) dx$

14. Compute $\int_0^{\frac{\pi}{4}} \tan^2(x) dx$

15. Compute $\int \sqrt{1 - x^2} dx$

16. Compute $\int \frac{x^3 + 2x^2 + 1}{x^4 + x^2} dx$

17. Compute $\int_0^1 xe^x \; dx$

18. Compute $\int \frac{2x + 1}{x^2 + x} dx$

19. Compute $\int x^3 \sqrt{x^2 + 1} dx$
20. Find the limit of the sequence $a_n = (1 - \frac{1}{n})^n$. Does the sequence $b_n = (-1)^na_n$ converge or diverge? Give a reason for your answer.
21. Write the repeating decimal $.\overline{5}$ (this means $.555555...$ no end to the $5 #39;s) as a geometric series. Use the geometric sum formula to find a rational number equal to this repeating decimal. Log in or sign up to see discussion or post a question. 22. Determine if the series converges or diverges. If it converges, find its sum. If it diverges, state why.$\sum_{n=0}^{\infty}(-1)^n \frac{3^n}{2^(n + 1)}$Log in or sign up to see discussion or post a question. 23. Apply the integral test to the series$\sum_{n=1}^{\infty} \frac{2}{n(n + 1)}$. The associated improper integral must be written and solved correctly. State conclusion obtained. Log in or sign up to see discussion or post a question. 24. Determine if the series converges or diverges. Give reasons for your answer.$\sum_{n=1}^{\infty} \frac{n^2 + n + 3}{2n^3 + 2n - 1}$. Log in or sign up to see discussion or post a question. 25. Determine if the series converges or diverges. Give reasons for your answer.$\sum_{n=1}^{\infty} \frac{1}{\sqrt{n^3}}$. Log in or sign up to see discussion or post a question. 26. Determine if the series converges or diverges. Give reasons for your answer.$\sum_{n=1}^{\infty} \frac{1}{2^n - 1}$. Log in or sign up to see discussion or post a question. 27. Determine if the series$\sum_{n=1}^{\infty} \sqrt[n]{n}$converges or diverges. Log in or sign up to see discussion or post a question. 28. Determine if the series$\sum_{n=1}^{\infty} \frac{(-1)^n n^2}{n!}$converges absolutely, converges conditionally or diverges. Log in or sign up to see discussion or post a question. 29. Determine if the series$\sum_{n=2} \frac{(-1)^n}{\ln n}$converges absolutely, converges conditionally or diverges. Log in or sign up to see discussion or post a question. 30. Determine the radius and interval of convergence of the power series$\sum_{n=1}^{\infty} \frac{3^n(x - 1)^n}{n + 1}$. Log in or sign up to see discussion or post a question. 31. Determine the radius of convergence of the power series$\sum_{n=1}^{\infty} \frac{n^n x^n}$. Log in or sign up to see discussion or post a question. 32. Determine the radius of convergence of the power series$\sum_{n=1}^{\infty} \frac{2^nx^n}{n!}$. Log in or sign up to see discussion or post a question. 33. Suppose that the power series$\sum_{n=0}^{\infty} a_nx^n$is convergent when$x = $and divergent when$x = 6$. Is the series convergent when$x = 3$? When$x = -7$? When$x = 5$? Explain. Log in or sign up to see discussion or post a question. 34. Use the geometric sum formula to find a power series, with radius of convergence, that converges to$f(x) = \frac{x}{2 + x}$. Log in or sign up to see discussion or post a question. 35. Find the$2^{nd}$degree Taylor polynomial of$f(x) = \sqrt[3]{x}$expanded at$a = 8$. Log in or sign up to see discussion or post a question. 36. Use the McLaurin series for$e^x$to find a power series converging to an antiderivative of$e^{-x^2}$. Log in or sign up to see discussion or post a question. 37. Let$\mathbf{u} = \mathbf{i} - \mathbf{j} + 2\mathbf{k}$and$\mathbf{v} = 2\mathbf{i} + \mathbf{j} + \mathbf{k}$. Find the vector projection of$\mathbf{u}$onto$\mathbf{v}$and the vector component of$\mathbf{u}$orthogonal to$\mathbf{v}$. Log in or sign up to see discussion or post a question. 38. Find a paramteric form of the line through the points$P = (1, 2, -1)$and$Q = (2, 1, 3)$. Log in or sign up to see discussion or post a question. 39. Find an equation of the plane containing the point$P = (1, 2, -1)$and the line$l(t) = (1 + t, 1 - 3t, 2 + t), -\infty < t < \infty$. Log in or sign up to see discussion or post a question. 40. Find the point of intersection of the line$l(t) = (1 + t, 1 - 3t, 2 + t), -\infty < t < \infty$with the plane$x + y + z = 1\$.