Practice 10 questions on ENGR233 - Applied Advanced Calculus (ENGR) at Concordia University. Free AI-generated quiz on uNotes — track your score, retake anytime.
1Find the equation of the plane that contains the point (1, 5, -1) and is perpendicular to the line of intersection of the planes $-x + y - 5z = 4$ and $2x - y - 2z = 0$.
2A thin wire has the shape of the curve C traced by $\vec{r}(t) = \cos(\pi t) \hat{i} + t \hat{j} + (\sin(\pi t) + 5) \hat{k}$ on the interval $0 \le t \le 1$. Find the length of the curved wire.
3Use Green's theorem to evaluate the line integral $\int_C (3x^2 - 4xy)dx + (y^3 - x^2)dy$ where C is the rectangle with vertices (2,1), (3,1), (3,6), and (2,6) traversed in a clockwise direction.
4Using spherical coordinates, find the volume of the solid inside $z = \sqrt{x^2 + y^2}$ and bounded by $z^2 + x^2 + y^2 = 1$ and $z^2 + x^2 + y^2 = 4$.
5Evaluate the surface integral $\iint_S y^3 z dS$ where S is the portion of the surface $x = 4 + z^2$ bounded by $z = 0, z = 1, y = 0$ and $y = 2$.
6Use the divergence theorem to compute the outward flux of the vector field $\vec{F} = 3x^2 \hat{i} + (2x + y) \hat{j} + (x^2 - z^2) \hat{k}$ through the finite cylinder $x^2 + y^2 = 4$, and $-1 \le z \le 3$.
7Use Stokes' Theorem to evaluate $\oint_C \vec{F} \cdot d\vec{r}$ where $\vec{F} = (4z + x) \hat{i} - 2xy \hat{j} + (x^2 - y) \hat{k}$ and C is a triangle in the first octant defined by the plane of $x + 3y + z = 3$ and C is oriented counter clockwise direction when viewed from above.
8Find the center of mass of the half cylinder whose shape is described by $x^2 + y^2 \le 4$, $x \ge 0$, and $-1 \le z \le 1$ and whose density is $\rho(x, y, z) = x^2$. Hint: Use and apply symmetry.
9Find the function $g(x,z)$ so that the vector $\vec{F} = (8x\cos(y)\sin^2(z) + 2ze^{2xz}) \hat{i} - 4x^2 \sin(y)\sin^2(z) \hat{j} + (g(x,z)\cos(y) + 2xe^{2xz}) \hat{k}$ is a conservative field.
10Use the change of variable technique to evaluate the integral $\iint_R (x^2 + y^2)(x^2 - y^2) dxdy$ where R is the region bounded by the graphs of $x = 0$, $x = 1$, $y = 0$, $y = 1$ by means of the change of variables $u = 2xy$ and $v = x^2 - y^2$.
