Calculus I · Unit 3B · answer key

Unit 3B Practice Exam B Answer Key

Finish an honest attempt first. Then compare one numbered response at a time, locate the first line where your reasoning diverged, and retry without the key open.

Answer 1

Problem 1

Intersections are $x=-1,2$ :

A=\int_{-1}^{2}(x+2-x^2)dx=\frac92.

Answer 2

Problem 2

V=\pi\int_0^1(1-x^2)^2dx=\frac{8\pi}{15}.

Answer 3

Problem 3

V=2\pi\int_0^1x(1-x^2)dx=\frac{\pi}{2}.

Answer 4

Problem 4

The area of a semicircle with diameter $d$ is $\pi d^2/8$ , so

V=\int_0^4\frac{\pi(2\sqrt{x})^2}{8}dx=4\pi.

Answer 5

Problem 5

S=2\pi\int_0^1x\sqrt{1+1^2}dx=\pi\sqrt2.

Answer 6

Problem 6

m=\int_0^1(1+x^2)dx=\frac43, \qquad \bar x=\frac{\int_0^1x(1+x^2)dx}{m}=\frac{9}{16}.

Answer 7

Problem 7

W=\int_{0.10}^{0.30}200x\,dx=8\text{ J}.

Answer 8

Problem 8

With $y$ measured from the bottom, slice volume is $\pi dy$ and lift distance is $3-y$ :

W=9800\pi\int_0^2(3-y)dy=39200\pi\text{ J}.

Answer 9

Problem 9

Let $y$ be depth below the surface. The width is $4(1-y/3)$ :

F=9800\int_0^3 y\,4\left(1-\frac y3\right)dy=58800\text{ N}.

Answer 10

Problem 10

C(300)=2000+\int_0^{300}(15+0.02q)dq=7400.

Answer 11

Problem 11

\int_0^1 3x^2dx=1, \qquad E[X]=\int_0^1x(3x^2)dx=\frac34.

Answer 12

Problem 12

Vertical strips are parallel to the $y$ -axis and create shells around the $y$ -axis; the same vertical strips are perpendicular to the $x$ -axis and create disks around the $x$ -axis. Both methods preserve the given $x$ -description without solving for an inverse function.

Answer 13

Problem 13

Units verify that the local contribution has the right physical dimension, while geometry determines radii, heights, widths, and distances. Ignoring either can yield a numerically evaluated integral that represents the wrong quantity.

After the explanation

Use the section idea

Reading lens

Mixed applications test modeling recognition: commit to a diagram and slice statement before looking for a familiar formula.

Mental model

A complete solution connects context, variable, bounds, slice contribution, integral, computation, units, interpretation, and a reasonableness check.

Decision

Classify the output and geometry first, solve from a blank start, then reveal one worked answer only to diagnose the earliest divergent decision.

Common trap

Reading a key before forming a setup or treating every wrong result as algebra when the real error was a radius, width, density, or bound.

Check yourself

Can you rebuild the setup without the answer open and defend every factor in a sentence?

Source & rights

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