MIT 18.01 Problem Set 7

This lesson reproduces the Problem Set “MIT 18.01 Problem Set 7” from MIT OpenCourseWare 18.01 Single Variable Calculus (Fall 2006) (https://ocw.mit.edu/courses/18-01-single-variable-calculus-fall-2006/), licensed under CC BY-NC-SA 4.0. Work through 5 problems in order: type your reasoning in the answer box or write it on the handwriting pad, then submit for AI grading.

How the grading works Every problem is graded by an AI mathematics teacher that reads the official solution shipped with the course. It checks your work criterion by criterion, awards partial credit for correct steps, and returns written feedback. Equivalent mathematics and valid alternative methods are accepted — a different route to the same result can still earn full marks.

If your answer is incomplete, the feedback names exactly which steps are missing, so you can correct them and try again.

(from PS6) The voltage $V$ of house current is given by

where $t$ is time, in seconds and $C$ is a constant amplitude. The square root of the average value of $V^2$ over one period of $V(t)$ (or cycle) is called the \emph{root-mean-square} voltage, abbreviated RMS. This is what the voltage meter on a house records. For house current, find the RMS in terms of the constant $C$. (The peak voltage delivered to the house is $\pm C$. The units of $V^2$ are square volts; when we take the square root again after averaging, the units become volts again.)

The solid torus is the figure obtained by rotating the disk $(x-b)^2+y^2 \leq a^2$ around the $y$-axis. Find its volume by the method of shells. (Hint: Substitute for $x-b$. As noted p. 229/11, the answer happens to be the area of the disk multiplied by the distance travelled by the center as it revolves.)

a) For any integer $n \geq 0$, use the substitution $\tan^2 x=\sec^2 x-1$ to show that

b) Deduce a formula for $\int \tan^4 x\,dx$.

a) Derive a formula for $\int \sec x\,dx$ by writing $\sec x=\frac{\cos x}{1-\sin^2 x}$ (verify this), and then making a substitution for $\sin x$ and using partial fractions. (Your final answer must be expressed in terms of $x$.)

b) Convert the formula into the more familiar one by multiplying the fraction in the answer on both top and bottom by $1+\sin x$. (Note that $(1/2) \ln u=\ln \sqrt{u}$.)

Find the volume under the first hump of the function $y=\cos x$ rotated around the $y$-axis by the method of shells.