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MIT 18.01 Exam 4

The graded exam on integration techniques: partial fractions, integration by parts with a logarithm, trigonometric substitution, arc length and polar area.

1. MIT 18.01 Exam 4 — how this assessment works

This lesson reproduces the Exam “MIT 18.01 Exam 4” 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 6 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.

2. MIT 18.01 EXAM4 Problem 1 — Partial fractions with a repeated linear factor (15 pts)

Evaluate

\int \frac{dx}{x(x+1)^2}.

3. MIT 18.01 EXAM4 Problem 2 — Integration by parts with a logarithm (15 pts)

Evaluate

\int (\ln x)\,x^2\,dx.

4. MIT 18.01 EXAM4 Problem 3 — Trigonometric substitution (20 pts)

Use a trigonometric substitution to evaluate

\int_0^1 \frac{dx}{(4+x^2)^{3/2}}

(Be careful evaluating the limits.)

5. MIT 18.01 EXAM4 Problem 4 — Arc length and surface area (10 pts + 10 pts)

a. (10 points) Find an integral formula for the arc length of the curve

y=2\sqrt{x+1}\quad\text{for}\quad 0\le x\le 1 .

Do not evaluate.

b. (10 points) Find an integral formula for the surface area of the curve in part (a) rotated around the $x$ -axis. Simplify the integrand and evaluate the integral.

6. MIT 18.01 EXAM4 Problem 5 — Spiral in polar coordinates: crossings and area (7 pts + 8 pts)

a. (7 points) Sketch the spiral

r=\theta^2,\qquad 0\le\theta\le 3\pi .

Say how many times the curve meets the $x$ -axis counting $\theta=0$ as the first time, and mark those points with X-s. (Your sketch need not be accurate to scale.)

b. (8 points) On your picture, shade in the region

0\le r\le\theta^2,\qquad 0\le\theta\le 2\pi,

and find its area.

7. MIT 18.01 EXAM4 Problem 6 — A line in polar coordinates (10 pts + 5 pts)

a. (10 points) Find the equation in polar coordinates for the line $y=x-1$ in the form

r=f(\theta).

b. (5 points) Find the range of $\theta$ for the portion of the line $y=x-1$ in the range $0\le x<\infty$ . (It helps to draw a picture.)

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