Calculus I · Unit 3B · answer key

Unit 3B Practice Exam A 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

A=\int_0^2(2x-x^2)dx=\frac43.

Answer 2

Problem 2

Intersections are $y=-1,2$ , and the line is to the right:

A=\int_{-1}^{2}\bigl[(y+2)-y^2\bigr]dy=\frac92.

Answer 3

Problem 3

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

Answer 4

Problem 4

V=2\pi\int_0^2x(4-x^2)dx=8\pi.

Answer 5

Problem 5

V=\int_0^2(2-x)^2dx=\frac83.

Answer 6

Problem 6

L=\int_0^3\sqrt{1+2^2}\,dx=3\sqrt5.

Answer 7

Problem 7

m=\int_0^3(2+x)dx=\frac{21}{2}, \qquad \bar x=\frac{\int_0^3x(2+x)dx}{m}=\frac{12}{7}\text{ m}.

Answer 8

Problem 8

W=\int_0^4(10+3x)dx=64\text{ J}.

Answer 9

Problem 9

Let $y$ measure height above the bottom. A slice has volume $12dy$ , weight $9800(12)dy$ , and lifting distance $3-y$ :

W=117600\int_0^2(3-y)dy=470400\text{ J}.

Answer 10

Problem 10

F=9800\int_0^3 2y\,dy=88200\text{ N}.

Answer 11

Problem 11

\int_{100}^{200}(20+0.04q)dq=2600\text{ dollars}.

Answer 12

Problem 12

P(X\le1/2)=\int_0^{1/2}2x\,dx=\frac14, \qquad E[X]=\int_0^1x(2x)dx=\frac23.

Answer 13

Problem 13

A correct response identifies, for example, $(\mathrm{N/m^3})(\mathrm{m^3})(\mathrm{m})=\mathrm{N\,m}=\mathrm{J}$ in a pumping problem. A reasonableness check may use sign, units, comparison with a constant-force estimate, geometric bounds, or expected magnitude.

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

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.