Wednesday, 2 September 2026

Evaluate $\int \sin (3x - 1) dx$.

\(\int_0^{\pi/4} \tan x \sec^2 x dx.\)

Find a fundamental set of Frobenius solutions of \[ x^2y extquotesingle extquotesingle{} - x(5-x)y extquotesingle{} + (9-4x)y = 0. \]

\(\lim_{y\to 0} \frac{\sin 2y}{3y}.\)

Calculate the centroid of the region between the curves $y = x$ and $y = \sqrt{x}$ with uniform density in the interval $0 \le x \le 1$.

Calculate the surface integral \(\iint_S (x-y)\,dS\), where \(S\) is the cylinder \(x^2+y^2=1\), \(0 \le z \le 2\), including the circular top and bottom.

Solve \(\frac{dy}{dx} = \sqrt{\frac{1-y^2}{1-x^2}}\).

Let $\mathcal{R}$ be the region bounded by the curve $y = \ln x$, the x-axis, and the line $x = e$. Find the area of $\mathcal{R}$.

Evaluate $\lim_{x \to 0} (\sin x)^{\cos x}$.

[T] Consider the torus of equation \(\left(x^{2}+y^{2}+z^{2}+R^{2}-r^{2}\right)^{2}=4 R^{2}\left(x^{2}+y^{2}\right)\), where $R \geq r > 0$. a. Write the equation of the torus in spherical coordinates. b. If $R=r$, the surface is called a horn torus. Show that the equation of a horn torus in spherical coordinates is $\rho=2 R \sin \varphi$. c. Use a CAS to graph the horn torus with $R=r=2$ in spherical coordinates.