← Calculus I

Unit 9: Applications of Integration

The definite integral at work: area between curves, solids built by slicing and revolving, motion from velocity, work against a force, exponential growth.

21 topics · 74 questions

Each topic is a path, not a menu: work down it in order and every question builds on the one above. Click any of them to watch the answer draw itself.

6.1 The Area Between Two Curves

  1. How it worksHow does slicing the region between two curves into thin vertical strips turn the area into a single integral?
  2. Common mistakeWhy isn't the area between two curves just the integral of the top function minus the integral of the bottom function taken separately?
  3. Worked exampleFind the area between f(x) = x2 - 4 and g(x) = 4 - x2.
  4. Worked exampleFind the area between f(x) = x2 and g(x) = x + 2 between their two points of intersection.
Your turn1 of 2
Find the area between f(x) = 7 - x2 and g(x) = x2 - 7.

6.1 The Area Between Curves That Cross More Than Once

  1. How it worksHow do you find every intersection point first so you know where the top and bottom functions trade places?
  2. Worked exampleFind the total area between f(x) = sin(x) and g(x) = cos(x) from x = 0 to x = π.
  3. Common mistakeWhy doesn't integrating (top minus bottom) across the whole interval work when the curves cross in the middle?
Your turn1 of 2
Find the total area between f(x) = 6sin(x) and g(x) = 6cos(x) from x = 0 to x = π.

6.1 Finding Area by Integrating with Respect to y

  1. How it worksHow does rewriting each curve as x in terms of y turn a sideways region into an ordinary area integral?
  2. Going furtherWhat happens to an area calculation when a region can't be described as a single function of x anywhere?
  3. Worked exampleFind the area enclosed by x = y2 and x = y + 2 by integrating with respect to y.
Your turn1 of 2
Find the area enclosed by x = y2 and x = 4y + 6 by integrating with respect to y.

6.2 Volumes by Slicing with General Cross-Sections

  1. How it worksHow does stacking infinitely many thin cross-sectional slabs turn a solid's volume into a single integral?
  2. Common mistakeWhy isn't the volume of a solid with cross-sections just the area of one cross-section multiplied by the total length?
  3. Worked exampleFind the volume of a solid whose base is bounded by y = x2 and y = 4, with square cross-sections perpendicular to the x-axis.
Your turn1 of 2
Find the volume of a solid whose base is bounded by y = 4x2 and y = 6, with square cross-sections perpendicular to the x-axis.

6.2 Volumes with Triangular and Semicircular Cross-Sections

  1. How it worksHow do you write the area of an equilateral triangle or a semicircle in terms of the same variable that describes the base?
  2. Common mistakeWhy doesn't a cross-section's shape need to match the shape of the solid's base at all?
  3. Worked exampleFind the volume of a solid whose base is the region under y = square root of x from 0 to 4, with semicircular cross-sections perpendicular to the x-axis.
Your turn1 of 2
Find the volume of a solid whose base is the region under y = 4·square root of x from 0 to 6, with semicircular cross-sections perpendicular to the x-axis (diameter equal to the region's height at each x).

6.2 The Disk Method

  1. How it worksWatch a two-dimensional region actually revolve around an axis and sweep out a three-dimensional solid.
  2. Worked exampleFind the volume of the solid formed by revolving y = square root of x around the x-axis from x = 0 to x = 4 using disks.
  3. Worked exampleFind the volume of the solid formed by revolving y = 4 - x2 around the x-axis from x = -2 to x = 2 using disks.
  4. Common mistakeWhy isn't the radius in the disk method always just the function's value, regardless of which axis you revolve around?
  5. Going furtherWhat happens to the disk method's setup when the region is revolved around the y-axis instead of the x-axis?
Your turn1 of 2
A region under y = 4·square root of x is revolved around the x-axis. Write the disk's cross-sectional area A(x), the integrand for the volume integral, as a function of x.

6.2 The Washer Method

  1. How it worksWatch a washer's hole appear as the inner curve is subtracted from the outer curve before the region revolves.
  2. Worked exampleFind the volume of the solid formed by revolving the region between y = x2 and y = 2x around the x-axis using washers.
  3. Going furtherWhat happens to the washer method's setup when the axis of revolution is a horizontal line other than the x-axis?
  4. Worked exampleFind the volume of the solid formed by revolving the region between y = square root of x and y = x/2 around the x-axis using washers.
  5. Common mistakeWhy doesn't the washer method reduce to the disk method's formula just by ignoring the inner radius entirely?
Your turn1 of 2
The region between y = x2 and y = 4x is revolved around the x-axis. Write the washer's cross-sectional area A(x), the integrand for the volume integral, as a function of x.

6.3 Volumes of Revolution: Cylindrical Shells

  1. How it worksWatch a single cylindrical shell unroll into a flat rectangle whose area gives the shell method its formula.
  2. Worked exampleFind the volume of the solid formed by revolving the region under y = x2 from x = 0 to x = 2 around the y-axis using shells.
  3. Common mistakeWhy isn't the shell method's radius measured from the curve itself instead of from the axis of revolution?
Your turn1 of 2
Find the volume of the solid formed by revolving the region under y = 4x2 from x = 0 to x = 4 around the y-axis using shells.

6.3 Choosing Disks, Washers, or Shells

  1. How it worksHow does the axis of revolution and the way the region is bounded decide whether disks, washers, or shells is the fastest method?
  2. Worked exampleBuild both a shell integral and a washer integral, without evaluating either, for revolving y = x2 from x = 0 to x = 2 around the y-axis.
  3. Going furtherWhat happens to the difficulty of a volume problem when the axis of revolution is moved from the y-axis to the line x = 3?
Your turn1 of 2
The region under y = 4x2 from x=0 to x=4 is revolved around the y-axis. Find its volume by evaluating a washer integral in terms of y (integrating from y=0 to y=4·42).

6.4 Arc Length of a Curve

  1. How it worksHow does the Pythagorean theorem applied to infinitesimally small steps along a curve build the arc length integral?
  2. Going furtherWhat happens to the arc length formula when the curve is described as x in terms of y instead of y in terms of x?
  3. Common mistakeWhy isn't the arc length of a curve just the straight-line distance between its two endpoints?
  4. Worked exampleFind the arc length of f(x) = x3/2 from x = 0 to x = 4.
Your turn1 of 2
For f(x) = 3x3/2, the arc-length integral is the integral of √(1+(f'(x))2) dx. Write this integrand as a function of x.

6.4 Surface Area of a Solid of Revolution

  1. How it worksHow does revolving a tiny piece of arc length around an axis sweep out a thin ring whose area you can add up?
  2. Worked exampleFind the surface area generated by revolving y = square root of x from x = 0 to x = 4 around the x-axis.
  3. Common mistakeWhy isn't the surface area of a solid of revolution the same calculation as its volume?
  4. Going furtherWhat happens to the surface area formula when the axis of revolution is the y-axis instead of the x-axis?
Your turn1 of 2
For f(x) = 3·square root of x, the surface-area integral is the integral of 2·π·f(x)*√(1+(f'(x))2) dx. Write this integrand as a function of x.

6.1 The Average Value of a Function

  1. How it worksHow does dividing a definite integral by the length of the interval produce a function's true average value?
  2. Common mistakeWhy isn't the average value of a function the same as the average of its value at the two endpoints?
  3. Worked exampleFind the average value of f(x) = x2 on the interval from 0 to 3.
Your turn1 of 2
Find the average value of f(x) = 4x2 + 6 on the interval from 0 to 8.

5.3 The Mean Value Theorem for Integrals

  1. How it worksHow does the Mean Value Theorem for Integrals guarantee a point where the function's height matches a rectangle of equal area?
  2. Worked exampleFind the value guaranteed by the Mean Value Theorem for Integrals for f(x) = x2 on the interval from 0 to 3.
  3. Common mistakeWhy doesn't the Mean Value Theorem for Integrals guarantee that point sits at the interval's midpoint?
Your turn1 of 2
For f(x) = 4x2 on the interval from 0 to 6, find the value c guaranteed by the Mean Value Theorem for Integrals, where f(c) equals the average value of f.

6.5 Recovering Position from Velocity

  1. How it worksHow does the Net Change Theorem applied to velocity recover an object's displacement over a time interval?
  2. Worked exampleFind the displacement of an object with velocity v(t) = t2 - 4t from t = 0 to t = 3.
  3. Common mistakeWhy isn't the displacement of an object always equal to the total distance it traveled?
Your turn1 of 2
An object has velocity v(t) = 4t2 - 6t and initial position s(0) = 4. Find its position function s(t).

6.5 Displacement Versus Total Distance Traveled

  1. How it worksHow does integrating the absolute value of velocity, instead of velocity itself, capture total distance traveled?
  2. Going furtherWhat happens to the total-distance integral when a velocity function never changes sign on the interval?
  3. Worked exampleFind the total distance traveled by an object with velocity v(t) = t2 - 4t from t = 0 to t = 3.
Your turn1 of 2
Find the total distance traveled by an object with velocity v(t) = t2 - 3t from t = 0 to t = 8.

6.5 Work Done by a Variable Force

  1. How it worksHow does integrating a changing force over distance turn Hooke's Law into a formula for the work stored in a spring?
  2. Worked exampleFind the work done stretching a spring 0.4 meters beyond its natural length if a force of 20 N stretches it 0.2 meters, using Hooke's Law.
  3. Worked exampleFind the work required to pump the water out of a cylindrical tank of radius 2 meters and height 6 meters through its top.
  4. Common mistakeWhy isn't the work done stretching a spring simply the final force multiplied by the total distance stretched?
Your turn1 of 2
A force of 22 N stretches a spring 0.36 meters beyond its natural length. Using Hooke's Law, find the work W(x) done stretching the spring x meters beyond its natural length.

6.5 Hydrostatic Force on a Submerged Plate

  1. How it worksHow does integrating pressure, which grows with depth, over a plate's area build up the total hydrostatic force?
  2. Worked exampleFind the hydrostatic force on a vertical rectangular plate 4 meters wide with its top edge 2 meters below the surface and its bottom edge 6 meters below.
  3. Going furtherWhat happens to the hydrostatic force on a plate when it is tilted instead of held vertically?
  4. Common mistakeWhy isn't the pressure on a submerged plate the same at every depth across its surface?
Your turn1 of 2
Find the hydrostatic force on a vertical rectangular plate 6 meters wide with its top edge 4 meters below the surface, as a function of the depth d of its bottom edge. Use a weight density for water of 9800 N per cubic meter.

6.6 Center of Mass of a Thin Rod or Flat Region

  1. How it worksHow does weighting each thin slice of a region by both its position and its mass locate the balance point?
  2. Common mistakeWhy isn't the center of mass of a region always the same as its geometric centroid when the density varies?
  3. Worked exampleFind the center of mass of a thin rod lying on the interval from 0 to 4 with density function rho(x) = x + 1.
  4. Going furtherWhat happens to a flat region's center of mass when all of its mass gets pushed toward one edge?
Your turn1 of 2
Find the center of mass of a thin rod on the interval from 0 to L with density function rho(x) = 4x + 4, as a function of the rod's length L.

6.8 Exponential Growth

  1. How it worksHow does solving the differential equation dy/dt equals ky produce the exponential growth formula y = y0·ekt?
  2. Common mistakeWhy isn't a quantity growing at 10 percent per year the same as a quantity growing by a constant amount each year?
  3. Worked exampleFind how long it takes a population of 500 bacteria growing at a continuous rate of 8 percent per hour to double.
Your turn1 of 2
A population of 727 grows continuously at a rate of 0.104 per hour. Find how long it takes to grow to 5 times its original size.

6.8 Exponential Decay and Half-Life

  1. How it worksHow does the same exponential model that drives growth flip its sign to drive decay and produce a half-life formula?
  2. Worked exampleFind the half-life of a substance that decays from 80 grams to 65 grams in 10 years.
  3. Worked exampleFind how much of a 200-gram sample with a half-life of 5 years remains after 15 years.
  4. Common mistakeWhy doesn't a substance with a half-life of 5 years ever actually reach zero mass?
Your turn1 of 2
A substance decays from 85 grams to 36 grams in 19 years. Find its half-life.

5.3 How the Accumulation Function Connects Back to Its Own Derivative

  1. How it worksWatch the accumulation function's slope at each point rise and fall to match the height of the integrand directly beneath it.
  2. Common mistakeWhy isn't a graph of the accumulation function simply a rescaled copy of the original integrand's graph?
  3. Ties togetherHow does connecting a function, its accumulation function, and its derivative graph tie together everything differentiation and integration do?
Your turn1 of 2
Let A(x) = the integral of (4t + 6) from 0 to x, the accumulation function for f(t)=4t+6. Find A'(x).