Second Midterm Midpoint Assessment

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General Concepts

If $f(x,y)$ is a continuous function on a region $D$ in the $xy$ -plane, then $\displaystyle \iint _D f(x,y)dA$ represents the signed areasigned volume of the region below the graph of $z = f(x,y)$ lying over the region $D$ .

If $f(x,y)$ is a continuous function and $C$ is a curve in the $xy$ -plane, then $\displaystyle \int _C f(x,y)ds$ is called a scalar line integralvector line integral and represents the total unsigned areasigned areasigned volume of the region under $z = f(x,y)$ lying over the curve $C$ .

Which of the following functions is an example of a vector field?

The temperature at any point $(x,y,z)$ in the air around us. The velocity of water at any point $(x,y,z)$ in a river. The air pressure at any point $(x,y,z)$ around us.

Give the definition of a conservative vector field.

What does a vector line integral $\displaystyle \int _C \textbf {F} \cdot d\textbf {r}$ represent geometrically?

If $R$ is a region in the $xy$ -plane, then what does $\displaystyle \iint _R 1dA$ represent?

If $C$ is a curve, then $\displaystyle \int _C 1ds$ gives what?

True/False Questions

Answer each of the following as true or false, and explain your answe. Unless otherwise specified, all functions and all components of vector fields are defined and continuous everywhere.

If $C$ is an oriented curve and $\textbf {F}$ is a vector field that, at every point on $C$ , makes an acute angle with the tangent vectors of $C$ , then $\displaystyle \int _C \textbf {F} \cdot dr > 0$ .

True False

If $f(x,y)$ is a continuous function and $C$ is a closed curve in the $xy$ -plane, then $\displaystyle \int _C f(x,y)ds = 0$ .

True False

The vector field $\textbf {F}(x,y) = \left \langle \sin (x^2),\sin (y^2) \right \rangle$ is conservative.

True False

If $\textbf {F} =\left \langle F_1,F_2 \right \rangle$ is a conservative vector field, then $\frac {\partial F_1}{\partial y} = \frac {\partial F_2}{\partial x}$ .

True False

If $C$ is the unit circle, oriented counterclockwise, and $\textbf {F}$ is a vector field with $\displaystyle \int _C \textbf {F} \cdot d\textbf {r} = 0$ , then $\textbf {F}$ is a conservative vector field.

True False

If $C$ is the unit circle, oriented counterclockwise, and $\textbf {F}$ is a vector field with $\displaystyle \int _C \textbf {F} \cdot d\textbf {r} = 0$ , then $\textbf {F}$ is perpendicular to the curve $C$ at every point on the circle.

True False

If $\textbf {F}$ is a conservative vector field, then $\textbf {F}$ has infinitely many potential functions $f$ .

True False

If $f(x,y)$ has continuous second partial derivatives on $\mathbb {R}^2$ , and $C_1$ and $C_2$ are two paths from $(0,0)$ to $(2,1)$ , then $\displaystyle \int _{C_1} \nabla f \cdot d\textbf {r} = \int _{C_2} \nabla f \cdot d\textbf {r}$ .

True False

If $C$ is an oriented curve and $C'$ is the curve $C$ with the opposite orientation, and $\textbf {F}$ is a vector field, then $\displaystyle \int _C \textbf {F} \cdot d\textbf {r}= \int _{C'} \textbf {F} \cdot d\textbf {r}$ .

True False

If $C$ is an oriented curve in the $xy$ -plane and $C'$ is the curve $C$ with the opposite orientation, and $f(x,y)$ is an everywhere continuous function, then $\displaystyle \int _C f(x,y)ds = \int _{C'} f(x,y)ds$ .

True False

If $\textbf {F} = \left \langle -y,0 \right \rangle$ , and $C$ is a closed curve (oriented counterclockwise) which encloses a region $R$ in the $xy$ -plane, then $\displaystyle \int _C \textbf {F} \cdot d\textbf {r} = \text {area}(R)$ .

True False

If $C$ is a curve from point $P$ to point $Q$ in the $xy$ -plane, and $f(x,y)$ is a continuous function, then $\displaystyle \int _C f(x,y)ds = f(Q) - f(P)$ .

True False

Free Response

If $f(x,y) = (x+2y)^2$ and $R$ is the rectangle $[0,1] \times [-1,1]$ in the $xy$ -plane, then $\displaystyle \iint _R f(x,y)dA = \answer {10/3}$

The volume of the solid bounded by $z = x^2+y^2$ and the planes $z = 0$ , $y = 0$ , $x=0$ , and $x+y=1$ is: $\answer {1/6}$

If $R$ is the region given by $y \leq \sqrt {1-x^2}$ , $y \geq x$ , and $y \geq -x$ , and $f(x,y) = y$ , then $\displaystyle \iint _R f(x,y)dA = \answer {\frac {\sqrt {2}}{3}}$ .

The volume of the solid bounded by $z = 8-x^2-y^2$ and $z = x^2+y^2$ is: $\answer {16 \pi }$ .

The integral $\int _0^4 \int _{\sqrt {y}}^2 \sin (x^3)dxdy = \answer {-\frac {1}{3}\cos (8)+\frac {1}{3}}$

If $C$ is the curve given by $\textbf {r}(t) = \left \langle 2t,3t,4t \right \rangle$ for $0 \leq t \leq 1$ , and $f(x,y,z) = z^2$ , then $\displaystyle \int _C f(x,y,z)ds = \answer {16\sqrt {29}/3}$ .

The integral $\displaystyle \int _{-1/\sqrt {2}}^0 \int _{-x}^{\sqrt {1-x^2}} (x-3)dydx$ in polar coordinates is: $\int _{\answer {\pi /2}}^{\answer {3\pi /4}} \int _{\answer {0}}^{\answer {1}} \answer {(r\cos (\theta )-3)r} dr\,d\theta ,$ and the integral is equal to: $\answer {\frac {1}{24}(-8+4\sqrt {2}-9\pi )}$ .

If $\textbf {F}(x,y) = \left \langle x+y,y-x \right \rangle$ and $C$ is the line segment from $(0,0)$ to $(1,2)$ , then $\displaystyle \int _C \textbf {F} \cdot d\textbf {r} = \answer {5/2}$ .

If $\textbf {F} = \left \langle y,x \right \rangle$ and $C$ is the line segment from $(1,-1)$ to $(2,-3)$ , then $\displaystyle \int _C \textbf {F} \cdot d\textbf {r} = \answer {-5}$ .

If $\textbf {F} = \left \langle x+y,y-x \right \rangle$ and $C$ is the circle $x^2+y^2=1$ oriented clockwise, then $\displaystyle \int _C \textbf {F} \cdot d\textbf {r} = \answer {2\pi }$ .

Let $R$ be the region in the first quadrant inside both the circle $x^2+y^2=1$ and the circle $(x-1)^2+y^2=1$ . Then $\displaystyle \iint _R ydA = \answer {5/24}$ .

If $\textbf {F} = \left \langle x^2,3x,xz \right \rangle$ and $C$ is the line segment from $(1,2,3)$ to $(1,0,4)$ , then $\int _C \textbf {F} \cdot d\textbf {r} = \answer {-5/2}$

Let $C$ be the cure going from $(0,0)$ to $(2,1)$ to $(5,1)$ to $(5,0)$ back to $(0,0)$ along straight line segments. Then $\int _C (x^3+e^{x+y}+e^{x^2})dx + (xy+e^{y^2}+e^{x+y})dy = \answer {-\frac {11}{6}}.$

If $C$ is the curve given by $\textbf {r}(t) = \left \langle 1+\sin ^3(t),1-\cos (t),t/\pi \right \rangle$ , with $0 \leq t \leq \pi /2$ , then $\int _C (yz+z\sin (xz))dx+(xz+y^3)dy+(xy+x\sin (xz))dz = \answer {\frac {9}{4}-\cos (1)}.$

Let $\textbf {F} = \displaystyle \left \langle x^2,e^{\cos (y)+y} \right \rangle$ and let $C$ be the arc of the parabola $y = 3x^2$ from $(1,3)$ to $(-1,3)$ . Then $\displaystyle \int _C \textbf {F} \cdot d\textbf {r} = \answer {-2/3}$ .

Let $C$ be the curve given by $\textbf {r}(t) = \left \langle \cos (t),t^2,\sin (t) \right \rangle$ for $-\pi \leq t \leq 0$ . Then $\displaystyle \int _C \sqrt {y}ds = \answer { \frac {1}{12}(1+4\pi ^2)^{3/2} - \frac {1}{12}}$ .

Let $C$ be the portion of $x^2+y^2=1$ where $x \geq 0$ , oriented from $(0,1)$ to $(0,-1)$ , and let $\textbf {F} = \left \langle y^2+y\sin (xy)-\sin (x^4),y^2+x\sin (xy) \right \rangle$ . Then $\int _C \textbf {F} \cdot d\textbf {r} = \answer {-\frac {2}{3}}$

If $C$ is the top half of $x^2+y^2=1$ oriented counterclockwise, then $\int _C \left (\cos (x)\cos (y)-\frac {y^2}{2}\right )dx + (2xy+\sin (y^6)-\sin (x)\sin (y))dy = \answer {2-2\sin (1)}.$