Calculus I · Unit 3B · answer key

Unit 3B Practice Exam B Answer Key

Review the worked solutions and modeling rubrics for Unit 3B Practice Exam B. Check methods, units, and interpretation.

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Section overview

Review, practice, exams, and answer keys

What this section is building

Review the worked solutions and modeling rubrics for Unit 3B Practice Exam B. Check methods, units, and interpretation.

Notice

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

Decide

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

Avoid

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.

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.

Source & rights

Original instruction with traceable references.

BetterGrades-original; no direct adaptation declared in the verified handoff.

Reference textbooks remain rights-separated and are not published as application assets. Any direct adaptation requires separate identification and attribution.

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

Answer 13

Problem 13

Source & rights

Original instruction with traceable references.

BetterGrades-original; no direct adaptation declared in the verified handoff.

Reference textbooks remain rights-separated and are not published as application assets. Any direct adaptation requires separate identification and attribution.

Unit 3B Practice Exam B Answer Key

Related resources

Original instruction with traceable references.

Unit 3B Practice Exam B Answer Key

Related resources

Original instruction with traceable references.