Calculus I · Unit 3B · practice

Unit 3B Cumulative Practice

Practice Unit 3B Cumulative Practice with cumulative integral-calculus problems designed to test setup, computation, units, and interpretation.

Supporting resourceUse this page when the main sequence reveals a specific gap or question.Open the Unit 3B map →

Section overview

Review, practice, exams, and answer keys

What this section is building

Practice Unit 3B Cumulative Practice with cumulative integral-calculus problems designed to test setup, computation, 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.

Practice method

Work in three passes

First, classify. Name the integral output and structural method before writing algebra. This separates a recognition error from a calculation error.

Second, solve without the key. Record a complete attempt, including bounds, constants, domains, units, or interpretation when the prompt asks for them.

Third, reveal one answer at a time. Compare the first line where your work differs, close the answer, and redo that item from a blank start.

Cumulative Practice

Exercise

Find the area enclosed by $y=x^2$ and $y=2x+3$ .

Answer reveal

Exercise 1 answer

Your setup or answer

Write a real attempt before opening the supplied answer.

Exercise

Rotate that same enclosed region about the horizontal line $y=-1$ . Set up and evaluate the washer integral.

Answer reveal

Exercise 2 answer

Your setup or answer

Write a real attempt before opening the supplied answer.

Exercise

Rotate that same enclosed region about the vertical line $x=-2$ . Set up and evaluate the shell integral.

Answer reveal

Exercise 3 answer

Your setup or answer

Write a real attempt before opening the supplied answer.

Exercise

The base is the region enclosed by $y=x^2$ and $y=2x+3$ . Cross-sections perpendicular to the $x$ -axis are squares. Find the volume.

Answer reveal

Exercise 4 answer

Your setup or answer

Write a real attempt before opening the supplied answer.

Exercise

Find the arc length of $y=x^{3/2}$ on $[0,4/9]$ .

Answer reveal

Exercise 5 answer

Your setup or answer

Write a real attempt before opening the supplied answer.

Exercise

A rod occupies $[0,1]$ and has density $\lambda(x)=1+x^2$ . Find its mass and center of mass.

Answer reveal

Exercise 6 answer

Your setup or answer

Write a real attempt before opening the supplied answer.

Exercise

A spring has constant $k=200\,\mathrm{N/m}$ . Find the work required to stretch it from $0.10\,\mathrm{m}$ to $0.30\,\mathrm{m}$ beyond equilibrium.

Answer reveal

Exercise 7 answer

Your setup or answer

Write a real attempt before opening the supplied answer.

Exercise

A $4\,\mathrm{m}$ -long triangular trough has ends $2\,\mathrm{m}$ deep and $3\,\mathrm{m}$ wide at the top. It is full of water. Set up and evaluate the work to pump the water to the top edge using $9800\,\mathrm{N/m^3}$ .

Answer reveal

Exercise 8 answer

Your setup or answer

Write a real attempt before opening the supplied answer.

Exercise

A triangular vertical gate has a $4\,\mathrm{m}$ top edge at the water surface and a vertex $3\,\mathrm{m}$ below. Find the hydrostatic force using $9800\,\mathrm{N/m^3}$ .

Answer reveal

Exercise 9 answer

Your setup or answer

Write a real attempt before opening the supplied answer.

Exercise

Marginal cost is $C'(q)=15+0.02q$ dollars per unit and fixed cost is $C(0)=2000$ . Find $C(300)$ .

Answer reveal

Exercise 10 answer

Your setup or answer

Write a real attempt before opening the supplied answer.

Exercise

A probability density has the form $p(x)=kx$ on $[0,2]$ . Find $k$ , verify normalization, and compute $E[X]$ .

Answer reveal

Exercise 11 answer

Your setup or answer

Write a real attempt before opening the supplied answer.

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.

Calculus I · Unit 3B · practice

Unit 3B Cumulative Practice

Practice Unit 3B Cumulative Practice with cumulative integral-calculus problems designed to test setup, computation, units, and interpretation.

Supporting resourceUse this page when the main sequence reveals a specific gap or question.Open the Unit 3B map →

Section overview

Review, practice, exams, and answer keys

What this section is building

Practice Unit 3B Cumulative Practice with cumulative integral-calculus problems designed to test setup, computation, 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.

Practice method

Work in three passes

First, classify. Name the integral output and structural method before writing algebra. This separates a recognition error from a calculation error.

Second, solve without the key. Record a complete attempt, including bounds, constants, domains, units, or interpretation when the prompt asks for them.

Third, reveal one answer at a time. Compare the first line where your work differs, close the answer, and redo that item from a blank start.

Cumulative Practice

Exercise

Find the area enclosed by $y=x^2$ and $y=2x+3$ .

Answer reveal

Exercise 1 answer

Your setup or answer

Write a real attempt before opening the supplied answer.

Exercise

Rotate that same enclosed region about the horizontal line $y=-1$ . Set up and evaluate the washer integral.

Answer reveal

Exercise 2 answer

Your setup or answer

Write a real attempt before opening the supplied answer.

Exercise

Rotate that same enclosed region about the vertical line $x=-2$ . Set up and evaluate the shell integral.

Answer reveal

Exercise 3 answer

Your setup or answer

Write a real attempt before opening the supplied answer.

Exercise

The base is the region enclosed by $y=x^2$ and $y=2x+3$ . Cross-sections perpendicular to the $x$ -axis are squares. Find the volume.

Answer reveal

Exercise 4 answer

Your setup or answer

Write a real attempt before opening the supplied answer.

Exercise

Find the arc length of $y=x^{3/2}$ on $[0,4/9]$ .

Answer reveal

Exercise 5 answer

Your setup or answer

Write a real attempt before opening the supplied answer.

Exercise

A rod occupies $[0,1]$ and has density $\lambda(x)=1+x^2$ . Find its mass and center of mass.

Answer reveal

Exercise 6 answer

Your setup or answer

Write a real attempt before opening the supplied answer.

Exercise

A spring has constant $k=200\,\mathrm{N/m}$ . Find the work required to stretch it from $0.10\,\mathrm{m}$ to $0.30\,\mathrm{m}$ beyond equilibrium.

Answer reveal

Exercise 7 answer

Your setup or answer

Write a real attempt before opening the supplied answer.

Exercise

Answer reveal

Exercise 8 answer

Your setup or answer

Write a real attempt before opening the supplied answer.

Exercise

A triangular vertical gate has a $4\,\mathrm{m}$ top edge at the water surface and a vertex $3\,\mathrm{m}$ below. Find the hydrostatic force using $9800\,\mathrm{N/m^3}$ .

Answer reveal

Exercise 9 answer

Your setup or answer

Write a real attempt before opening the supplied answer.

Exercise

Marginal cost is $C'(q)=15+0.02q$ dollars per unit and fixed cost is $C(0)=2000$ . Find $C(300)$ .

Answer reveal

Exercise 10 answer

Your setup or answer

Write a real attempt before opening the supplied answer.

Exercise

A probability density has the form $p(x)=kx$ on $[0,2]$ . Find $k$ , verify normalization, and compute $E[X]$ .

Answer reveal

Exercise 11 answer

Your setup or answer

Write a real attempt before opening the supplied answer.

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.