Mr. Integral
Math notes/Calculus II/Parametric and Polar Calculus

Calculus II Checkpoint

3 MIN READ · PRACTISE AT YOUR OWN PACE   ·   FREE

LESSON 3 OF 3See the unit map ↗

Choose a method, state its conditions and check the result across integration, series and curves.

Builds on Polar Curves, Area and Length

The bigger question: What if x is not the most useful input?

On this page

How to use this checkpoint

Attempt each problem before opening the solutions. Write the method and its conditions as well as the final value. If you can calculate but cannot explain why a method applies, revisit the corresponding unit before moving on to multivariable calculus.

Use four checks throughout: Are the bounds attached to the current variable? Is the quantity signed or geometric? Does the convergence test actually decide this case? Does the parameter interval trace the object once?

Visual guide

VISUAL GUIDEChoose the representation that matches the region
A radius-2 circle can be parametrized by (2 cos t, 2 sin t), described by x² + y² = 4, or written in polar form r = 2. The shaded quarter disk has area π; these descriptions encode the same geometry for different calculus tasks.-2.5-2.5-1.25-1.25001.251.252.52.5xy
  • Circle of radius 2
A radius-2 circle can be parametrized by (2 cos t, 2 sin t), described by x² + y² = 4, or written in polar form r = 2. The shaded quarter disk has area π; these descriptions encode the same geometry for different calculus tasks.

Problems

  1. Evaluate ∫01xln⁡(1+x)dx\int_0^1x\ln(1+x)dx.
  2. Determine whether ∫0∞dx/(1+x)2\int_0^\infty dx/(1+x)^2 converges, and evaluate it.
  3. Classify ∑n=1∞(−1)n−1/n\sum_{n=1}^\infty(-1)^{n-1}/n as absolutely convergent, conditionally convergent or divergent. Give an error bound after NN terms.
  4. Find the full interval of convergence of ∑n=1∞(x−1)n/n2\sum_{n=1}^\infty(x-1)^n/n^2.
  5. Approximate e0.1e^{0.1} with a quadratic Maclaurin polynomial and give a valid error bound.
  6. Rotate the region 0≤y≤x0\le y\le\sqrt x, 0≤x≤10\le x\le1, about the xx-axis. Find its volume.
  7. Find the distance traveled by (x,y)=(3cos⁡t,3sin⁡t)(x,y)=(3\cos t,3\sin t) for 0≤t≤π0\le t\le\pi.
  8. A Simpson calculation uses five equal subintervals. Explain what must change before applying the composite Simpson formula in this course.

Worked solutions

Show worked solutions
  1. Use parts with u=ln⁡(1+x)u=\ln(1+x) and dv=xdxdv=x dx. The integral is 12ln⁡2−12∫01x2/(1+x)dx\tfrac12\ln2-\tfrac12\int_0^1x^2/(1+x)dx. Divide: x2/(1+x)=x−1+1/(1+x)x^2/(1+x)=x-1+1/(1+x). The logarithms cancel and the result is 1/41/4.
  2. Use a finite upper limit first: [−1/(1+x)]0R=1−1/(1+R)→1[-1/(1+x)]_0^R=1-1/(1+R)\to1. The integrand is bounded at zero and only the infinite tail is improper.
  3. It converges by the alternating test, but the absolute harmonic series diverges. Convergence is conditional, and ∣RN∣≤1/(N+1)|R_N|\le1/(N+1).
  4. The ratio test gives radius 11 around 11. At both endpoints x=0,2x=0,2, absolute values give 1/n21/n^2, so the interval is [0,2][0,2].
  5. 1+0.1+0.12/2=1.1051+0.1+0.1^2/2=1.105. The third derivative on [0,0.1][0,0.1] is at most e0.1e^{0.1}, giving error at most e0.1(0.1)3/6<0.000185e^{0.1}(0.1)^3/6<0.000185.
  6. Disks have radius x\sqrt x, so V=π∫01xdx=π/2V=\pi\int_0^1x dx=\pi/2.
  7. Speed is 33, so distance is 3π3\pi. Displacement magnitude is 66 and is a different quantity.
  8. Use an even number of subintervals, such as six, and recompute the equally spaced samples and weights. Do not silently apply the alternating 4,24,2 pattern to an odd count.
PAUSE & THINKA quick check, not a grade

Try it yourself.

Choose an answer and explain your reasoning to yourself. Use a hint if you get stuck.

Which reasoning is sufficient to show Σₙ₌₁^∞ 1/n² converges?

Hint 1 · Find a starting point

A necessary condition alone is not a convergence proof.

Hint 2 · Take the next step

The ratio test is inconclusive at 1, but the p-series criterion applies.

Show the reasoning

Answer: It is a p-series with p=2>1.

The p-series test proves convergence. Terms tending to zero do not suffice, as the harmonic series shows.

Before moving on: what would make one of the other answers wrong? Saying why is part of understanding.

Choose the next step

If integration methods or bounds were difficult, return to the first three units. If series classification or endpoints were difficult, revisit the convergence and power-series units. If geometry was difficult, draw the slices or parameter interval before repeating the calculation. Calculus III reuses these habits with more than one variable; Linear Algebra can be studied alongside it.

MAKE IT YOURS

Pause before the next idea.

Can you explain how the visualization connects to this lesson’s goal? If a step still feels uncertain, put this lesson on your review list and try the check again another day.

Optional marks, not a grade. Saved in this browser only. Open notebook →