MIT 18.01 Practice Quiz 4A

This lesson reproduces the Practice Quiz “MIT 18.01 Practice Quiz 4A” 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 9 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.

Evaluate

Evaluate

by making the substitution $x=2\tan u$.

a) Derive a reduction formula relating

to

b) Let $F(x)=\int_0^1 e^{-x^2}\,dx$. Express $\int_0^1 x^2e^{-x^2}\,dx$ in terms of values of $F(x)$.

Find the volume of the solid obtained by rotating about the $y$-axis the finite region bounded by the positive $x$-and $y$-axes and the graph of $y=\cos x$.

Make a reasonable sketch of one loop of the polar curve $r=\sin 3\theta$, and find the area inside it.

Let $x(t)=\cos^3 t$, $y(t)=\sin^3 t$, $0\le t\le\pi/2$ be a parametric representation of a curve.

a) Compute the arclength of the curve.

b) Compute the surface area of the surface formed by rotating the curve around the $x$-axis.

Set up an integral for the length of one arch of the curve $y=\sin x$, and by estimating the integral, tell how this length compares with $\pi\sqrt{2}$.

A circular metal disc of radius $a$ has a non-constant density $\delta$ (units: gms/cm$^2$); the density at a point $P$ on the disc is given by $\delta=r^2$, where $r$ is the distance of the point from the center of the disc. Set up and evaluate a definite integral giving the total mass of the disc.

a) Sketch the curve given in polar coordinates by $r=1+\cos\theta$.

b) Find the polar coordinates of the following two points (show work):

(i) where the curve in part (a) intersects the circle of radius $3/2$ centered at the origin;

(ii) where the above curve intersects the circle of radius $3/2$ centered at the point $x=3/2$ on the $x$-axis.