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Exercise 1: Definite Integrals (Recap)

A

Determine an antiderivative (an indefinite integral) for each of the functions

$$x^3\,,\,\,\frac{1}{x^3}\,\,\,\,\mathrm{and}\,\,\,\,\sin\left(3x-\frac{\pi}{2}\right)\,.$$

B

Compute the following definite integrals:

$$\int_0^{1}x^3\,\mathrm{d}x\,,\,\,\int_1^{2}\frac{1}{x^3}\,\mathrm{d}x\,\,\,\,\mathrm{and}\,\,\,\,\int_{-\frac{\pi}{2}}^{0}\sin\left(3x-\frac{\pi}{2}\right)\,\mathrm{d}x\,.$$

Exercise 2: Integration by Parts. By Hand

A

Determine an antiderivative for the function $\,x\cos(x)\,,$ and then do a check to ensure that your result is correct.

B

Determine the indefinite integral $\displaystyle{\,\int{t\e^t \,\mathrm dt}\,},$ and then check your result.

C

Determine an antiderivative of the function $\,x^2\ln(x)\,,\,\,x>0\,.$

D

A first-order linear differential equation is given by

$$\,x'(t)-2x(t)=3t\,.$$

Solve it using the general solution formula.

Exercise 3: Integration by Substitution

For the questions in this Exercise, use the substitution method via the formula:

$$\int{f(g(x))g'(x)\,\mathrm dx}=\int{f(t)\,\mathrm dt}\,\,\,\mathrm{where}\,\,\,t=g(x)\,.$$
A

Determine an antiderivative of

$$\,\displaystyle{h(x)=x\e^{x^2}}\,.$$

B

Compute the indefinite integral

$$\displaystyle{\int \frac{x}{x^2+1} \, \mathrm dx\,.}$$

C

Compute

$$\displaystyle{\int_0^{\pi} \frac{\sin (x)}{3 -\cos(x)} \mathrm dx\,.}$$

Exercise 4: Parametrization and a Line Integral. By Hand

If a curve in the $(x,y)$ plane is given as the graph of a function

$$y=f(x)\,,\,\,x\in \left[a,b\right]\,,$$

then it is easy to state a parametric representation of the curve:

$$\begin{matr}{c}x\newline y\end{matr}=\mathbf r(u)=\begin{matr}{c}u\newline f(u)\end{matr}\,,\,\,u\in \left[a,b\right]\,.$$
A

A curve $K$ is given as a segment of the graph of the function $\ln(x)\,$:

$$K=\left\lbrace (x,y)\in\reel^2\,\vert\, y=\ln(x)\,,\,\,x\in\left[ 1\,,\,2\sqrt 2\right] \right\rbrace .$$

State a parametric representation of the curve and determine the Jacobian that belongs to the parametric representation.

B

Compute the line integral

$$\displaystyle{\int_Kx^2\,\mathrm d\mu\,.}$$

Exercise 5: Area and Volume. Advanced

Please enjoy the glass skyscraber The Gherkin by architect Norman Forster, located in London, UK.

gherkin.jpg gherkinMaple.png

In order to model the skyscraper, we begin by delimiting in the $\,(x,z)$ plane a region $\,A\,$ by the coordinate axes and the graph of the function

$$x=f(z)=\frac 12\,\sqrt{-z^2+2z+3}\,,\,\,\,\,z\in\left[0,3\right]\,$$

according to the graph above.

A

A solid model of The Gherkin appears when we rotate region $\,A\,$ about the $\,z$-axis by an angle of $\,2\pi\,.$ Determine the volume of this model.

B

Determine the area of region $\,A\,.$