#### MATH 53

##### Final | Summer '12| Quilodran
(a) (10 points) Let $C$ be the boundary of the region enclosed by the parabola $y=x^2$ and the line $y=1$ with counter clockwise orientation. Calculate $\int_C (y^2 + e^{\sqrt{x}}) dx + x\; dy$.

(b) (10 points) If the directional derivatives at the point $(1,1)$ are given Find Let $S$ be the surface parameterized by where the domain of the parameters is $D = { (u, v) | 0 \le u \le \frac{\pi}{2}, 0 \le v \le \sin^2 u }$.
(a) (10 points) Find the tangent plane at the point (b) (10 points) Calculate (20 points) define Compute $\int_C \mathbf{F} \cdot d\mathbf{r}$, where $C$ is the line segment from $(1,2,4)$ to $(1,1,1)$.
Hint: Calculate the line integrals for $\mathbf{G}$ and $\mathbf{H}$ separately. Use a different method for each integral.

(20 points) Let $S$ be the ellipsoid of equation $x^2 + \frac{y^2}{2} + \frac{z^3}{3} = 1$ and let $(u,v,w)$ be a point in $S$ with $u >0, v >0$ and $w >0$.
The tangent plane to $S$ at (u,v,w) has equation and together with the three coordinate planes encloses a (pyramid-like) solid $E$ whose volume equals Find the point $(u,v,w)$ as in the first paragraph such that $E$ has the minimum possible volume. Write what that volume is .

5. (20 points) Let $E$ be the solid enclosed by the paraboloids and let $S$ be the boundary of $E$ with outward pointing normal. Calculate where Simplify your answer.

Let $C$ be the curve consisting of: a line segment from $(0,0,0)$ to $(1,0,1)$ followed by the arc of a circle $x= \cos t, y= \sin t$, followed by the line segment from $(0,1,1)$ to $(0,0,0)$
(a) (5 points) Parametrize the two line segments (with the stated orientations) and verify that $C$ lies in the cone of equation (b) (15 points) Calculate 7. (20 points) Let $g$ be a function of one variable such that the derivatives are contineous on R. Define that is,f(x,y) equals the second derivativeof $g$ evaluated at For the disc Calculate in terms of the values of at 0 and 3.