Calculus Practice

Applications of Integrals Practice

Practice area, volume, average value, accumulated change, displacement, total distance, and other applications of definite integrals.

Start Practicing Review Applications

Try each problem first, then click Show solution to check your setup and reasoning.

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Applications Essentials

Area

\[ A = \int (\text{top}-\text{bottom}) \,dx \]

With vertical slices, subtract bottom from top.

Washers

\[ V = \pi \int (R^2-r^2) \,dx \]

Outer disk minus inner disk.

Shells

\[ V = 2\pi \int (\text{radius}) (\text{height}) \,dx \]

Radius times height times circumference factor \(2\pi\).

Download Practice PDF Review Applications

Level 1

Area and Signed Area

Practice area under curves, signed area, and total geometric area.

Problem 1

Area Under a Curve

Find the area under \[ y=x^2 \] from \(x=0\) to \(x=2\).

Step-by-Step Solution

Because the function is above the \(x\)-axis,

\[ A = \int_0^2x^2\,dx. \]
\[ = \left[ \frac{x^3}{3} \right]_0^2 \] \[ = \frac83 \]
Answer: \[ \boxed{ \frac83 } \]

Problem 2

Signed Area

Suppose \[ \int_0^2f(x)\,dx=5 \] and \[ \int_2^4f(x)\,dx=-3. \] Find \[ \int_0^4f(x)\,dx. \]

\(8\)
\(2\)
\(-2\)

Step-by-Step Solution

\[ \int_0^4f(x)\,dx = \int_0^2f(x)\,dx + \int_2^4f(x)\,dx. \]
\[ = 5+(-3) = 2 \]
Answer: \[ \boxed{2} \]

Problem 3

Total Area

The graph of \(f\) is above the \(x\)-axis from \(x=0\) to \(x=2\) with area \(6\), and below the \(x\)-axis from \(x=2\) to \(x=5\) with geometric area \(4\). What is the total geometric area?

\(2\)
\(10\)
\(-10\)

Step-by-Step Solution

Total geometric area counts both regions positively.

\[ 6+4=10 \]
Answer: \[ \boxed{10} \]

Level 2

Area Between Curves

Identify bounds, determine which function is on top, and build the correct area integral.

Problem 4

Intersections

Find the intersection points of \[ y=2x \] and \[ y=x^2. \]

Step-by-Step Solution

Set the equations equal:

\[ 2x=x^2. \] \[ x(x-2)=0. \]
Answer: \[ \boxed{ x=0,\;2 } \]

Problem 5

Area Between Curves

Find the area enclosed by \[ y=2x \] and \[ y=x^2. \]

Step-by-Step Solution

The curves intersect at \(x=0\) and \(x=2\).

On this interval, \(2x\) is above \(x^2\).

\[ A = \int_0^2 (2x-x^2)\,dx. \]
\[ = \left[ x^2-\frac{x^3}{3} \right]_0^2 \] \[ = 4-\frac83 = \frac43 \]
Answer: \[ \boxed{ \frac43 } \]

Problem 6

Horizontal Slices

When integrating with respect to \(y\), which subtraction order is used for area?

Top minus bottom
Right minus left
Left minus right

Step-by-Step Solution

A horizontal slice extends from the left boundary to the right boundary.

Answer: Right minus left.

Level 3

Volumes of Revolution

Practice disks, washers, and cylindrical shells.

Problem 7

Disk Method

The region under \[ y=x \] from \(x=0\) to \(x=2\) is rotated about the \(x\)-axis. Find the volume.

Step-by-Step Solution

The radius is

\[ R(x)=x. \]

Use the disk formula:

\[ V = \pi \int_0^2x^2\,dx. \]
\[ = \pi \left[ \frac{x^3}{3} \right]_0^2 \] \[ = \frac{8\pi}{3} \]
Answer: \[ \boxed{ \frac{8\pi}{3} } \]

Problem 8

Washer Method

A region has outer radius \[ R(x)=3 \] and inner radius \[ r(x)=x \] for \(0\leq x\leq 2\). Write and evaluate the washer-method volume.

Step-by-Step Solution

\[ V = \pi \int_0^2 \left( 3^2-x^2 \right) dx. \]
\[ = \pi \left[ 9x-\frac{x^3}{3} \right]_0^2 \] \[ = \pi \left( 18-\frac83 \right) \] \[ = \frac{46\pi}{3} \]
Answer: \[ \boxed{ \frac{46\pi}{3} } \]

Problem 9

Shell Method

The region under \[ y=x \] from \(x=0\) to \(x=2\) is rotated about the \(y\)-axis. Use cylindrical shells to find the volume.

Step-by-Step Solution

Radius:

\[ r=x. \]

Height:

\[ h=x. \] \[ V = 2\pi \int_0^2x(x)\,dx. \]
\[ = 2\pi \int_0^2x^2\,dx \] \[ = 2\pi \left[ \frac{x^3}{3} \right]_0^2 \] \[ = \frac{16\pi}{3} \]
Answer: \[ \boxed{ \frac{16\pi}{3} } \]

Level 4

Average Value and Accumulation

Use integrals to calculate average values and accumulated quantities.

Problem 10

Average Value

Find the average value of \[ f(x)=x^2 \] on \([0,2]\).

Step-by-Step Solution

\[ f_{\text{avg}} = \frac{1}{2-0} \int_0^2x^2\,dx. \]
\[ = \frac12 \left( \frac83 \right) = \frac43 \]
Answer: \[ \boxed{ \frac43 } \]

Problem 11

Net Change

Water enters a tank at the rate \[ r(t)=4t+2 \] liters per minute. How much water enters from \(t=0\) to \(t=3\)?

Step-by-Step Solution

Integrate the rate over the time interval.

\[ \int_0^3(4t+2)\,dt. \]
\[ = \left[ 2t^2+2t \right]_0^3 \] \[ = 18+6 = 24 \]
Answer: \(24\) liters.

Problem 12

Accumulation Function

Let \[ A(x) = \int_1^x (t^2+3)\,dt. \] Find \(A'(x)\).

Step-by-Step Solution

By the Fundamental Theorem of Calculus,

\[ \frac{d}{dx} \left[ \int_a^xf(t)\,dt \right] = f(x). \]
Answer: \[ \boxed{ A'(x)=x^2+3 } \]

Level 5

Motion and Mixed Applications

Distinguish displacement from distance and choose the correct integral setup.

Problem 13

Displacement

An object has velocity \[ v(t)=2t-4 \] for \(0\leq t\leq4\). Find its displacement.

Step-by-Step Solution

\[ \text{Displacement} = \int_0^4(2t-4)\,dt. \]
\[ = \left[ t^2-4t \right]_0^4 \] \[ = 16-16 = 0 \]
Answer: \[ \boxed{0} \]

Problem 14

Total Distance

For the same velocity \[ v(t)=2t-4, \] find the total distance traveled from \(t=0\) to \(t=4\).

Step-by-Step Solution

Velocity changes sign when

\[ 2t-4=0 \Rightarrow t=2. \]

Split the interval:

\[ \text{Distance} = - \int_0^2(2t-4)\,dt + \int_2^4(2t-4)\,dt. \]
\[ = 4+4 = 8 \]
Answer: \[ \boxed{8} \]

Problem 15

Method Selection

A region is rotated around the \(y\)-axis. Using vertical slices, which volume method naturally produces cylindrical shells?

Disk method
Washer method
Shell method

Step-by-Step Solution

Vertical slices are parallel to the \(y\)-axis. Rotating them around the \(y\)-axis creates cylindrical shells.

Answer: Shell method.

Problem 16

Mixed Review

Which expression gives the average value of a continuous function \(f\) on \([a,b]\)?

\[ \int_a^bf(x)\,dx \]
\[ \frac{1}{b-a} \int_a^bf(x)\,dx \]
\[ \frac{f(a)+f(b)}{2} \]

Step-by-Step Solution

Divide the accumulated value by the length of the interval.

Answer: \[ \boxed{ \frac{1}{b-a} \int_a^bf(x)\,dx } \]

Before You Finish

Applications of Integrals Checklist

  1. Distinguish signed area from total geometric area.
  2. Find intersection points before setting up area-between-curves problems.
  3. Use top minus bottom for vertical slices.
  4. Use right minus left for horizontal slices.
  5. Identify disk radii correctly.
  6. Distinguish outer and inner radii for washers.
  7. Use \(2\pi(\text{radius})(\text{height})\) for shells.
  8. Divide by \(b-a\) when finding average value.
  9. Interpret the integral of a rate as net change.
  10. Distinguish displacement from total distance.
  11. Split intervals where velocity changes sign when finding distance.
  12. Check units in application problems.

Continue Learning

Related Calculus Resources

Review the concepts behind these applications or continue through the Calculus learning path.

Applications Guide

Review area, volume, average value, accumulation, and motion.

Integrals Guide

Review antiderivatives, definite integrals, and the Fundamental Theorem.

Integrals Practice

Practice the integration skills used throughout this page.

Calculus Learning Hub

Browse the complete Calculus learning path.

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