重积分练习题

Question 1 (10 marks)

Find the equation of the tangent plane to $4 x^2-9 y^2-9 z^2-36=0$ at point $(3 \sqrt{3}, 2,2)$

Question 2 ( 18 marks)

Given that $ K =\left\|\frac{d T(t)}{d s}\right\|$ and $\mathbf{T}(t)=\frac{\mathbf{r}^{\prime}(t)}{\left\|\mathbf{r}^{\prime}(t)\right\|}=\frac{d \mathbf{r}(t) / d t}{d s(t) / d t}$, show that $K=\frac{\left\|\mathbf{r}^{\prime}(t) \times \mathbf{r}^{\prime \prime}(t)\right\|^{\prime}}{\left\|\mathbf{r}^{\prime}(t)\right\|^3}$

Question 3 ( 8 marks)

Find the unit tangent vector and the curvature at any point on the given curve.

$$ \bar{r}(t)=\left\langle e^t \cos t, e^t \sin t, t\right\rangle $$

Question 4 (8 marks)

Evaluate $\iint_R\left(x^2+y^2\right) d x d y, \mathrm{R}$ is the quarter-circle: $0 \leq y \leq \sqrt{1-x^2}, 0 \leq x \leq 1$

Question 5 (14 marks)

Evaluate $\iint_R\left(x^2+y^2\right) d x d y, \mathrm{R}$ is the triangle with vertices $(0,0),(1,0),(1,1)$.

Question 6 (14 marks)

Evaluate $\iint_{R_y}(x+y)^3 d x d y$. The $R_{x y}$ can be converted to $u=x+y, v=x-2 y$ where $1 \leq u \leq 4,-2 \leq v \leq 1$

Question 7 (16 marks)

Transform the following integral to cylindrical coordinates: $\int_0^1 \int_0^{\sqrt{1-x^2}} \int_0^{1+x+y}\left(x^2-y^2\right) d z d y d x$

Question 8 (16 marks)

Transform the following integral to spherical coordinates: $\iiint_{R_{y x}} x^2 y d x d y d z$, where $R_{x y z}$ is the sphere:

$$ x^2+y^2+z^2 \leq a^2 $$