MIT 18.01 Problem Set 8B

This lesson reproduces the Problem Set “MIT 18.01 Problem Set 8B” 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 2 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.

(Lecs 34-36, 6+5 pts: $1+2+2+1$, plus 5 extra)

a) Use L'Hospital's rule to evaluate

b) Use part (a) and limit comparison to show that the improper integral $\int_0^\infty x^n e^{-x}\,dx$ converges for $n \geq 0$. (Do not integrate by parts.)

c) Denote $A(n)=\int_0^\infty x^n e^{-x}\,dx$. Use integration by parts to find the constant $c_n$ for which

(Explain what happens at the infinite limit using part (a).)

d) Find $A(0)$ and deduce from part (c) the formula for $A(n)$, $n=0, 1, 2, \dots$.

e) (optional; 5pts extra credit) Show that the improper integral representing $A(-1/2)$ converges and evaluate it using a change of variables.

(Review problem using solids and surfaces of revolution; 9pts: $4+2+3$)

a) Show that the volume $V$ of the solid of revolution enclosed on the top by a spherical cap of radius $r$ of height $h$ and underneath by a horizontal plane is

Draw a picture. Doublecheck this answer in the three cases: the empty set ($h=0$); the half ball ($h=r$); the whole ball ($h=2r$). We will always assume that $0 \leq h \leq 2r$. Why?

b) Find the surface area of the spherical cap of radius $r$ and height $h$. Doublecheck your answer by evaluating it in the three cases $h=0$, $h=r$ and $h=2r$.

c) We will now find the shape of a soap bubble sitting on a table. Suppose that the bubble is a portion of a sphere as above, that it encloses a fixed volume $V$, and that its surface area is as small as possible (area of the curved surface, not counting the flat bottom=table). Find the minimizing shape, and describe it geometrically. (Hint: find $h/r$.)