Calculus I · Unit 3B · practice

Physics Application Studio

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.

Physics Application Studio

Modeling lab

A compact set of standard models

For each situation, identify the local contribution, variable of integration, bounds, units, and final interpretation before evaluating.

Exercise

A particle has velocity $v(t)=(t-1)(t-3)$ for $0\le t\le4$ . Find its displacement and total distance traveled, and explain how the sign changes affect the two totals.

Answer reveal

Exercise 1 answer

Your setup or answer

Write a real attempt before opening the supplied answer.

Exercise

A spring requires $12\,\mathrm{N}$ of force at an extension of $0.06\,\mathrm{m}$ . Find the work required to stretch it from $0.10\,\mathrm{m}$ to $0.25\,\mathrm{m}$ beyond equilibrium.

Answer reveal

Exercise 2 answer

Your setup or answer

Write a real attempt before opening the supplied answer.

Exercise

A rectangular tank is $4\,\mathrm{m}$ long, $3\,\mathrm{m}$ wide, and $2\,\mathrm{m}$ deep. It is full of water, which is pumped to an outlet $1\,\mathrm{m}$ above the top. Using water weight density $9800\,\mathrm{N/m^3}$ , set up and evaluate the work.

Answer reveal

Exercise 3 answer

Your setup or answer

Write a real attempt before opening the supplied answer.

Exercise

A triangular vertical plate has a $4\,\mathrm{m}$ top edge at the water surface and a vertex $3\,\mathrm{m}$ below the surface. Using water weight density $9800\,\mathrm{N/m^3}$ , find the hydrostatic force.

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Exercise 4 answer

Your setup or answer

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Exercise

A rod occupies $0\le x\le L$ and has linear density $\lambda(x)=a+bx$ , where $a,b,L>0$ . Find its mass, first moment about the origin, and center of mass.

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Exercise 5 answer

Your setup or answer

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Exercise

Force-position data are $F(0)=5$ , $F(1)=7$ , $F(2)=8$ , $F(3)=10$ , and $F(4)=14$ newtons. Estimate the work from $x=0$ to $x=4$ meters with the Trapezoidal Rule and state the units.

Answer reveal

Exercise 6 answer

Your setup or answer

Write a real attempt before opening the supplied answer.

After the explanation

Use the section idea

Reading lens

Translate the situation into rate, density, force, pressure, or probability before calculating; the integral is the final accumulation step, not the first modeling decision.

Mental model

Work adds force through distance, pumping adds slice weight through lift distance, pressure adds depth-dependent strip force, and marginal or probability models add weighted local contributions.

Decision

Draw a coordinate system, define the slice at a general position, express every changing factor in one variable, and state the domain and units.

Common trap

Confusing mass density with weight density, measuring depth from the wrong reference, or integrating a marginal quantity without an initial value when a total function is requested.

Check yourself

Does the setup respond correctly when the slice moves, and can a units or scale check expose a missing factor?

Source & rights

Original instruction with traceable references.

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