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MIT 18.01 Exam 3

The graded exam on the definite integral: substitution, interval approximations, volumes of revolution and functions defined by integrals.

1. MIT 18.01 Exam 3 — how this assessment works

This lesson reproduces the Exam “MIT 18.01 Exam 3” 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.

2. MIT 18.01 EXAM3 Problem 1 — Integrals by substitution (20 pts)

Evaluate the following integrals.

\int_0^2 \frac{x\,dx}{(1+x^2)^2}

\int_{-\pi/2}^{\pi/2} \sin^6 x \cos x\,dx

3. MIT 18.01 EXAM3 Problem 2 — Two-interval approximations (20 pts.)

Find the following approximations to

\int_0^{\pi/2} \cos x\,dx

(Do not give a numerical approximation to square roots; leave them alone.)

a) Using the upper Riemann sum with two intervals

b) Using the trapezoidal rule with two intervals

c) Using Simpson's rule with two intervals

4. MIT 18.01 EXAM3 Problem 3 — Volume of a solid of revolution (20 points)

Find the volume of the solid of revolution formed by revolving the $y$ -axis the region enclosed by

y=\cos(x^2)

and the $x$ -axis (central hump, only).

5. MIT 18.01 EXAM3 Problem 4 — Sleep and exam scores (20 points)

Students studying for an exam get $x$ hours of sleep in the two days leading up to the exam, where $x$ is the range $0 \leq x \leq a$ . The numbers of students who got between $x_1$ and $x_2$ hours of sleep in given by

\int_{x_1}^{x_2} c\,x\,dx; \quad 0 \leq x_1 \leq x_2 \leq a

a) What fraction o the student got less than $a/2$ hours of sleep?

b) Their scores are proportional to the amount of sleep they got: $S(x)=100(x/a)$ . Find the (correctly weighted) average score in the class.

6. MIT 18.01 EXAM3 Problem 5 — Accumulation function and change of variable (20 points)

Let

F(x)=\int_0^x \sqrt{t}\,\sin t\,dt

a) Find $F'(x)$ for $x > 0$ identify the points $a > 0$ $F'(a)=0$

b) Decide whether $F$ has a local maximum at the smallest critical point $a > 0$ that you found in part (a) by evaluating $F''$ .

c) Say whether $F(x)$ is positive, negative or zero at each of the following points, and give a reason in each case.

i) $x=0$

ii) $x=\pi$

iii) $x=2\pi$

d) Use a change of the variable to express $G(x)=\int_0^x u^2\sin(u^2)\,du$ in terms of $F$ .

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