Integrals

Surface Integrals, Divergence, Curl and Potential Exercise 1 : Coordinate Planes Given three squares with an area of 4 units each. They are placed: in the x-yplane, in the x-zplane, and in the y-zplane. Determine the surface elements. Exercise 2 : Rectangle Vector Element Given a rectangle with area $a \cdot b$, determine the vector element. Exercise 3 : Flux Through Given Surface Elements Compute the flow of the vector field $\mathbf{F}(x, y, z) = (5, 3, 0)$ through the surfaces given by the respective surface elements: ...

Multiple Integrals; Coordinate Systems Exercise 1 : Multiple Integrals with Constant Limits Evaluate the following multiple integrals: $\int_{y=0}^{b} \int_{x=0}^{a} dx dy$ $\int_{y=0}^{2} \int_{x=0}^{1} x^2 dx dy$ $\int_{x=0}^{\pi} \int_{y=0}^{\pi} \sin x \sin y dx dy$ $\int_{n=1}^{2} \int_{v=2}^{4} n(1+v) dv dn$ $\int_{x=-1/2}^{1/2} \int_{y=-1}^{1} \int_{z=0}^{2} dx dy dz$ $\int_{x=0}^{1} \int_{y=0}^{y_1} \int_{z=0}^{z_1} e^{az} dx dy dz$ Exercise 2 : Multiple Integrals with Variable Limits Evaluate the integrals: $\int_{x=0}^{2} \int_{y=x-1}^{3x} x^2 dy dx$ $\int_{x=0}^{1} \int_{y=0}^{2x} \int_{z=0}^{x+y} dx dy dz$ Using a double integral, obtain the area of an ellipse and the position of the center of mass of the half ellipse $(x \geq 0)$. The equation of an ellipse is $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$. Exercise 3 : Coordinate Systems A point has Cartesian coordinates $P = (3, 3)$. What are its polar coordinates? Give the equation of a circle of radius $R$ in Cartesian coordinates and polar coordinates. Obtain the equation of the spiral shown in Fig. 13.22 in polar coordinates. Evaluate $\int_{\theta=0}^{\pi/4} \int_{r=0}^{a} r^2 \cos \theta , dr , d\theta$. Exercise 4 : Cylindrical Coordinates Compute the volume of the hollow cylinder shown in Fig. 13.23 using cylindrical coordinates. Evaluate the volume of a cone of radius $R$ and height $h$. Obtain the moment of inertia of the cone about its center axis. The density $\rho$ is a constant. Exercise 5 : Moment of Inertia Calculate the moment of inertia of a sphere of radius $R$ and of constant density $\rho$ about an axis through its center, using spherical coordinates.

Integral Calculus Exercise 1 : Primitives Find the primitives of the following functions and the value of the constant: $f(x) = 3x$ given $F(1) = 2$ $f(x) = 2x + 3$ given $F(1) = 0$ Exercise 2 : Definite Integrals (Cosine) Evaluate the following definite integrals: $\int_{0}^{\pi/2} 3 \cos x , dx$ $\int_{-\pi/2}^{\pi/2} 3 \cos x , dx$ $\int_{0}^{\pi} 3 \cos x , dx$ Exercise 3 : Absolute Areas Obtain the absolute values of the areas corresponding to the following integrals: ...

Line, Volume and Surface Integrals Exercise 1 : Basic Line Integral For $\mathbf{v} = x^2,\hat{\mathbf{x}} + 2yz,\hat{\mathbf{y}} + y^2,\hat{\mathbf{z}}$, compute $\int_C \mathbf{v} \cdot d\mathbf{l}$ from $(0,0,0)$ to $(1,1,1)$ along: $(0,0,0) \to (1,0,0) \to (1,1,0) \to (1,1,1)$ $(0,0,0) \to (0,0,1) \to (0,1,1) \to (1,1,1)$ The straight line $x = y = z$ Evaluate $\oint \mathbf{v} \cdot d\mathbf{l}$ around the closed loop: out via 1., back via 2.. Exercise 2 : Gradient Field Verification Given $\phi = e^{xyz}$, compute $\nabla \phi$ and verify the gradient theorem: $\int_{\mathbf{a}}^{\mathbf{b}} \nabla \phi \cdot d\mathbf{l} = \phi(\mathbf{b}) - \phi(\mathbf{a})$ for $\mathbf{a} = (0,0,0)$ and $\mathbf{b} = (1,1,1)$ along two different paths. ...

Integrals Exercise 1 : Proofs Prove $\int_{0}^{b} x^{3} dx = \frac{b^4}{4}$ using equal partitions and $\sum_{i=1}^{n} i^3$. Prove $\int_{0}^{b} x^{4} dx = \frac{b^5}{5}$ analogously. Show $\lim_{n \to \infty} \sum_{k=1}^{n} \frac{k^p}{n^{p+1}} = \frac{1}{p+1}$. Prove $\int_{0}^{b} x^p dx = \frac{b^{p+1}}{p+1}$. Exercise 2 For $0 < a < b$, find $\int_{a}^{b} x^p dx$ using partitions with fixed ratios $r = t_i/t_{i-1}$: Show $t_i = a \cdot c^{i/n}$ where $c = b/a$. For $f(x) = x^p$, derive: $$ U(f,P) = (b^{p+1} - a^{p+1}) \frac{c^{p/n}}{1 + c^{1/n} + \cdots + c^{p/n}} $$ and find $L(f,P)$. Conclude $\int_{a}^{b} x^p dx = \frac{b^{p+1} - a^{p+1}}{p+1}$. Exercise 3 Evaluate by symmetry: ...