Calculus II, Fall 1998, Exam 1

1.

(35 pts total) Determine whether each of the following improper integrals converges or diverges. If it converges, find its value.

(a) [15 pts] $$\int^4_2 \frac{2}{x^2 - 4} \,dx$$

(b) [20 pts] $$\int^{\infty}_{1} 2x^3e^{-x^2} \,dx$$

2.

(30 pts) The region R is bounded by the x-axis and the curve $y = \cos\,x$ for $-\pi/2~\leq~x~\leq~\pi/2$. Determine the coordinates of the centroid $(\bar{x},\bar{y})$ of R.

HINT: Think before you integrate!

ANS. $\bar{x} = {\underline{\hspace*{1.0in}}},\; \bar{y} = {\underline{\hspace*{1.0in}}}$

3.

(35 pts total) The curve below is known as a ``swallowtail", with representation x = 12t - 4t³, y = -6t² + 3t⁴, $- 1.8 \leq t \leq 1.8$.
NOTE: The function for y differs slightly from that given out in class, so the function looks different.

(a) [6 pts] Find the two values of t corresponding to the point of intersection.

ANS. $t = {\underline{\hspace*{1.0in}}},\; {\underline{\hspace*{1.0in}}}$

(b) [12 pts] Find the slope of either tangent line to the curve at the point of intersection. [Also give the value of t for the tangent line whose slope you found.]

ANS. $\mbox{slope} = {\underline{\hspace*{1.0in}}}\; \mbox{for}\; t = \,{\underline{\hspace*{1.0in}}}$

(c) [5 pts] Show the direction of increasing t on the curve.

(d) [12 pts] Denote the values of t that correspond to the points A and B by t_A and t_B, respectively. Set up an integral, in terms of t, t_A, and t_B, for the length of the arc AB, the ``trailing edge" of the swallowtail. [Simplify the integral as much as possible, but you need not evaluate the integral or find the numbers t_A and t_B.]

EXTRA CREDIT [5 pts] Find the values of t_A and t_B.

ANS. $t_A = {\underline{\hspace*{1.0in}}}\; t_B = \,{\underline{\hspace*{1.0in}}}$

4.

(30 pts total)

[8pt] (a) The curves $r = f_1(\theta) = (1 + \sqrt{2})\cos\theta$ and $r = f_2(\theta) = 1 + \cos\theta$ for $-\pi < \theta \leq \pi$ are shown above.

(i) [4 pts] Write in the small boxes which curve is f₁ and which is f₂.

(ii) [6 pts] Circle the points of intersection of f₁ and f₂ on the graph. Find $(r,\theta)$ coordinates for all these points.

(iii) [8 pts] On the graph ``shade" the area inside f₁ and outside f₂, and write an expression involving one or more integrals for this area. [Do NOT simplify or evaluate the expression.]