Applications of single and several variable calculus (MTH112M 26-27)

Term
Semester I
Duration
Sep 22, 2026 - present
Instructor
Debasis Sen and Abhijit Pal
TA
Indrajit Ghosh

Room: L18

Class Timing: Tue 10-10:50

Course Description: Website

Office Hours: Wed 10:00 - 10:50 (Google Meet )

Textbooks and References

  1. George B. Thomas Jr., Maurice D. Weir & Joel R. Hass., Thomas Calculus, (15th edition).
  1. Robert G. Bartle & Donald R. Sherbert., Introduction to Real Analysis.

Lectures

Disclaimer: There are two versions of notes provided for each lecture. The Live Note is what I write on the board during class, while the Typeset Note (LaTeX) is the version I prepare beforehand to have on hand during the lecture, so it may be a bit more elaborate. Both are for reference only.

Lecture 1
Date: Sep 22, 2026

Assignments

Assignment 1: Polar Coordinates, Applications of Integrations (Click to view)
Original PDF
  1. Find the area of the region in the first quadrant bounded on the left by the \(y\)-axis, below by the curve \(x = 2\sqrt{y}\), above left by the curve \(x = (y - 1)^2\), and above right by the line \(x = 3 - y\).
  2. Sketch the graph of the following curves, given in polar coordinates, in the Cartesian plane:
    \[ r = \frac{\sin \theta}{2}, \quad r^2 = -\cos \theta, \quad r = e^{\theta/10}, \quad r = \frac{1}{2} + \sin \theta, \quad r = -|\cos \theta|. \]
    3. Angle θ
    Polar Curve
    Live θ Value:
    0.00 π
    Interval: [0, 2π]
  3. Find the area of the region that is inside the circle \(r = 3\sin\theta\) and also inside \(r = 1 + \sin\theta\).
  4. Find the intersection points between the curves \(r = \cos 2\theta\) and \(r = \sin 2\theta\).
  5. Find the lengths of the polar curves:
    1. \(r(\theta) = \frac{6}{1 + \cos\theta}, \quad 0 \leq \theta \leq \frac{\pi}{2}\),
    2. \(r(\theta) = \frac{a\sin^2\theta}{2}, \quad 0 \leq \theta \leq \pi, \quad a > 0\).
  6. (Non-rectifiable curve) Let \(g : [0, 1] \to [0, 1]\) be defined by
    \[ g(t) = \begin{cases} t \cos \left(\frac{\pi}{2t}\right), & t \neq 0, \\ 0, & t = 0. \end{cases} \]
    Show that \(g\) is continuous. Consider the curve \(\alpha : [0, 1] \longrightarrow \mathbb{R}^2, \alpha(t) = (t, g(t))\). Show that the curve is non-rectifiable i.e., it has infinite length.
    Hint: Recall that the length of a continuous curve \(\alpha : [0, 1] \to \mathbb{R}^2\) is defined by
    \[ L(\alpha) = \sup_{0 = s_0 < s_1 < \cdots < s_m = 1} \sum_{j=0}^{m-1} ||\alpha(s_j) - \alpha(s_{j+1})||, \quad (1) \]
    where the supremum is taken over all possible partitions \(0 = s_0 < s_1 < \cdots < s_m = 1\) of the interval \([0, 1]\) (no bound on \(m\)). For our specific example, take the partitions
    \[ P_n = \left\{0, \frac{1}{2n}, \frac{1}{2n - 1}, \ldots, \frac{1}{3}, \frac{1}{2}, 1\right\}, \quad n \in \mathbb{N}. \]
    Let \(\ell_n\) denote the length of the polygonal path inscribed in the curve using this partition. Prove that
    \[ \ell_n > \sum_{k=1}^{2n} \frac{1}{k} \]
    and deduce that the curve \(\alpha\) is not rectifiable.
  7. Consider the curve \(C\) defined by \(x(t) = \cos^3 t, y(t) = \sin^3 t, \quad 0 \leq t \leq \frac{\pi}{2}\).
    1. Find the length of the curve.
    2. Find the area of the surface generated by revolving \(C\) about the \(x\)-axis.
      3. 3D Surface of Revolution: Astroid about x-axis
      Sweep Angle φ:
      0.00 π
      Interval: [0, 2π]
  8. Let \(C\) denote the circular disc of radius \(b\) centered at \((a, 0)\) where \(0 < b < a\). Find the volume of the torus that is generated by revolving \(C\) around the \(y\)-axis using
    1. the Washer Method
    2. the Shell Method.
    9. 3D Torus Volume: Washer vs Shell Method
    Sweep Progress:
    y = -0.80
    Interval: y ∈ [-0.8, 0.8]
  9. A curved wedge is cut from a right circular cylinder of radius 3 by two planes. One plane is perpendicular to the axis of the cylinder and passes through the centre; the second plane also passes through the centre and meets the first plane at a \(45^\circ\) angle. Find the volume of the wedge (the region of the cylinder between the two planes).
    10. 3D Slicing Method: Volume of a Cylindrical Wedge

    This visualization matches your lecture notes. It shows the static, transparent wedge while the slicing right triangle sweeps across the \(x\)-axis to accumulate the volume.

    Cross-Section Slice:
    x = -3.00
    Interval: x ∈ [-3, 3]
  10. Using the theorem of Pappus, find the centroid of a thin, flat plate covering the region enclosed by the parabolas \(y = 2x^2\) and \(y = 3 - x^2\).
Assignment 2: Vectors, Curves, Surfaces, Vector Equations (Click to view)
Original PDF
  1. Consider the planes
    \[ x + y + z = 1, \] \[ 2x + ay + 3z = 4, \] \[ x + 2y + 2z = b, \]
    where \(a\) and \(b\) are parameters. Determine the values of \(a\) and \(b\) such that the three planes
    1. intersect at a single point,
    2. intersect in a line,
    3. intersect (taken two at a time) in three distinct parallel lines.
  2. Evaluate the distance between the lines
    \[ \frac{x - 2}{4} = \frac{y - 7}{-4} = \frac{z + 2}{3} \quad \text{and} \quad \frac{x + 1}{1} = \frac{y + 2}{4} = \frac{z - 1}{-3}. \]
  3. Show that the distance between the point \(Q = (x_0, y_0, z_0)\) and the plane
    \[ (x, y, z) \cdot \mathbf{n} = d \]
    is given by
    \[ \frac{|\mathbf{n} \cdot (x_0, y_0, z_0) - d|}{||\mathbf{n}||}. \]
    1. Find an equation for the surface consisting of all points that are equidistant from the point \((-1, 0, 0)\) and the plane \(x = 1\).
    2. Find an equation for the cylinder generated by a line through the curve
      \[ x^2 + y^2 = 4x, \quad z = 0 \]
      moving parallel to the vector \(\mathbf{i} + \mathbf{j} + \mathbf{k}\).
  5. Find the arc-length re-parametrization and then compute the curvature of the curve \(\alpha(t) = (e^t \cos t, e^t \sin t)\), \(t \in \mathbb{R}\).
  6. Let \(\alpha : (a, b) \to \mathbb{R}^2\) be a plane differentiable curve such that \(\alpha'(t) \neq 0\) for all \(t \in (a, b)\) and all its normals pass through a fixed point in \(\mathbb{R}^2\). Show that the image of \(\alpha\) lies on a circle.
  7. Show that the parabola \(y = ax^2\), \(a \neq 0\) has its largest curvature at its vertex and has no minimum curvature.
  8. Find the point on the curve \(c(t) = (5 \sin t)\vec{i} + (5 \cos t)\vec{j} + (12t)\vec{k}\) at a distance \(26\pi\) units from \((0, 5, 0)\) along the curve in the direction of increasing arc length.

Last Updated: Sep 27, 2026