← Calculus III (Multivariable)

Unit 4: Multiple Integration

A double integral is a volume under a surface, re-swept the other way, mapped through polar grids, then stretched by a Jacobian before tripling into solids.

26 topics · 94 questions

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5.1 Double Integrals over Rectangular Regions

  1. How it worksHow does a Riemann sum over a rectangular grid turn into a double integral as the grid gets finer?
  2. Worked exampleBuild the double Riemann sum estimate for the volume under z = x + y over the rectangle [0,2] x [0,2] using a 2x2 grid.
  3. Common mistakeWhy doesn't a double integral require you to pick an order before you can add up the little box volumes?
Your turn1 of 2
Find the double integral of x + 4y over the rectangle [0,4] x [0,5].

5.1 A Double Integral as the Volume Under a Surface

  1. How it worksHow does slicing the solid under a surface into thin slabs turn a volume problem into an iterated integral?
  2. Worked exampleFind the volume under z = 4 - x2 - y2 over the rectangle [0,1] x [0,1].
  3. Going furtherWhat happens to this volume interpretation once the surface dips below the xy-plane somewhere in the region?
  4. Common mistakeWhy doesn't a negative value for a double integral mean the volume it represents is somehow negative?
Your turn1 of 2
Find the volume under z = 8 - x2 - y2 over the rectangle [0,1.47] x [0,1.9].

5.1 Iterated Integrals and Fubini's Theorem

  1. How it worksHow does Fubini's theorem let you swap the order of an iterated integral over a rectangle without changing the answer?
  2. Worked exampleFind the iterated integral of x2 y over y from 0 to 2 and x from 0 to 3.
  3. Worked exampleFind the same integral of x2 y over the same rectangle, integrating with respect to x first instead.
  4. Common mistakeWhy doesn't Fubini's theorem guarantee you can swap the order of integration once the region isn't a plain rectangle?
Your turn1 of 2
Find the iterated integral of 4x2y over y from 0 to 5 and x from 0 to 4.

5.2 Double Integrals over General Regions

  1. How it worksHow do the inner integral's limits become functions of the outer variable when a region has curved boundaries?
  2. Worked exampleFind the double integral of xy over the region bounded by y = x2 and y = x for x between 0 and 1.
  3. Common mistakeWhy can't a double integral's limits always be written as four fixed numbers?
  4. Worked exampleFind the double integral of x over the region bounded by y = 0, y = x2, and x = 2.
Your turn1 of 2
Find the double integral of xy over the region bounded by y = x2 and y = 3x for x between 0 and 3.

5.2 Type I and Type II Regions

  1. How it worksHow does a Type I region stack vertical strips while a Type II region stacks horizontal ones?
  2. Worked exampleSketch and describe the region bounded by y = x2 and y = 4 as both a Type I and a Type II region.
  3. Going furtherWhat happens when a region isn't cleanly Type I or Type II on its own?
  4. Worked exampleFind the double integral of 1 over the region between y = x and y = x2 by treating it as a Type I region.
Your turn1 of 2
Find the double integral of 1 over the region between y = x and y = 4x2 (for x between 0 and 1/4), treating it as a Type I region.

5.2 Switching the Order of Integration

  1. Common mistakeWhy doesn't switching the order of integration mean you can just swap the numbers in the limits without re-deriving them?
  2. How it worksHow does re-sweeping the same region with horizontal strips instead of vertical ones change the limits of integration?
  3. Worked exampleFind the switched order and evaluate the integral of sin(y2) over y from x to 1 and x from 0 to 1.
  4. Worked exampleSketch the region for the integral of f(x,y) over y from x to 1 and x from 0 to 1, then switch its order.
Your turn1 of 2
Find the value of the integral of sin(2.24y2) over y from x to 1 and x from 0 to 1, by switching the order of integration.

5.2 Area of a Planar Region by Double Integration

  1. How it worksHow does the double integral of 1 over a region collapse into the same area formula a single integral would give?
  2. Worked exampleFind the area of the region bounded by y = x2 and y = 2x, using a double integral.
  3. Going furtherWhat changes about finding area by double integration once a region is bounded by three curves instead of two?
Your turn1 of 2
Find the area of the region bounded by y = 3x2 and y = 4x, using a double integral.

5.3 Double Integrals in Polar Coordinates

  1. How it worksHow does an extra factor of r appear in dA once a double integral switches to polar coordinates?
  2. Worked exampleFind the double integral of x2 + y2 over the disk of radius 3 centered at the origin, using polar coordinates.
  3. Common mistakeWhy can't you convert a double integral to polar coordinates just by replacing x with r cos(theta) and y with r sin(theta) alone?
Your turn1 of 2
Converting the double integral of x2 + y2 to polar coordinates turns x2+y2 into r2, and dA into r dr dtheta. Write the resulting full integrand r2·r, as a function of r.

5.3 Converting a Region to Polar Coordinates for Integration

  1. Worked exampleFind the polar limits for the region between x2 + y2 = 1 and x2 + y2 = 4 in the first quadrant.
  2. Worked exampleFind the double integral of e−(x^2+y^2) over the entire plane, using polar coordinates.
  3. Going furtherWhat happens to the polar limits of integration once a region is a cardioid instead of a circle?
  4. Common mistakeWhy doesn't every circular-looking region convert into a rectangle in the r-theta plane?
Your turn1 of 2
Find the double integral of e−2.94(x^2+y^2) over the entire plane, using polar coordinates.

5.2 Surface Area of a Graph z = f(x, y)

  1. How it worksHow do the two partial derivatives of f combine into the stretch factor inside a surface area integral?
  2. Worked exampleFind the surface area of the part of the plane z = 2x + 2y + 1 that lies above the unit square [0,1] x [0,1].
  3. Worked exampleFind the surface area of the part of the paraboloid z = x2 + y2 that lies below the plane z = 4.
  4. Going furtherWhat happens to the surface area integral's stretch factor as a surface gets steeper and steeper?
Your turn1 of 2
Find the surface area of the part of the plane z = 4x + 4y + 4 that lies above the rectangle [0,1.2] x [0,0.66].

5.2 Average Value of a Function Over a Region

  1. How it worksHow does dividing a double integral by a region's area turn a total into an average value?
  2. Worked exampleFind the average value of f(x, y) = x + y over the rectangle [0,2] x [0,4].
  3. Common mistakeWhy isn't the average value of a function over a region just the average of its values at the corners?
  4. Going furtherWhat happens to a function's average value over a region as that region shrinks down to a single point?
Your turn1 of 2
Find the average value of f(x, y) = x + 4y over the rectangle [0,4] x [0,6].

5.6 Mass of a Planar Lamina from Density

  1. How it worksHow does integrating a density function over a region add up mass the same way integrating 1 adds up area?
  2. Worked exampleFind the mass of the lamina occupying the square [0,1] x [0,1] with density rho(x,y) = x + y.
  3. Going furtherWhat happens to the mass integral once the density function is constant across the whole lamina?
  4. Common mistakeWhy doesn't doubling a lamina's density function automatically double the location of its center of mass?
Your turn1 of 2
Find the mass of the lamina occupying the rectangle [0,4] x [0,2] with density rho(x,y) = 4x + 4y.

5.6 Moments and the Center of Mass

  1. How it worksHow do the moments about the x- and y-axes get divided by the total mass to locate the center of mass?
  2. Worked exampleFind the center of mass of the lamina on [0,2] x [0,1] with density rho(x,y) = x.
  3. Common mistakeWhy isn't the center of mass of a lamina always the same as its plain geometric center?
Your turn1 of 2
Find the x-coordinate of the center of mass of the lamina on [0,4] x [0,6] with density rho(x,y) = 4x.

5.6 Moments of Inertia

  1. How it worksHow does moment of inertia weight each bit of mass by the square of its distance from the axis?
  2. Worked exampleFind the moment of inertia about the x-axis of the lamina on [0,1] x [0,1] with density rho(x,y) = 1.
  3. Ties togetherHow does the moment of inertia integral extend the same weighting idea used to find the center of mass?
Your turn1 of 2
Find the moment of inertia about the x-axis of the lamina on [0,4] x [0,5] with constant density rho(x,y) = 4.

5.6 Symmetry Shortcuts in Center of Mass Calculations

  1. How it worksHow does an odd density function paired with a symmetric region force one of the moment integrals to vanish?
  2. Worked exampleFind the x-coordinate of the center of mass of a semicircular lamina of radius 2 with constant density, using symmetry.
  3. Common mistakeWhy doesn't a symmetric region alone guarantee a shortcut if the density function isn't symmetric too?
Your turn1 of 2
Find the x-coordinate of the center of mass of a semicircular lamina of radius 7 (flat side along the y-axis, bulging into x > 0) with constant density, using symmetry.

5.6 Center of Mass and Moments of a Solid Region

  1. How it worksHow do the moments about the three coordinate planes get divided by total mass to locate a solid's center of mass?
  2. Worked exampleFind the center of mass of the solid box [0,2] x [0,1] x [0,1] with density rho(x,y,z) = x.
  3. Going furtherWhat happens to a solid's center of mass calculation once its density is constant throughout?
Your turn1 of 2
Find the x-coordinate of the center of mass of the solid box [0,4] x [0,6] x [0,2] with density rho(x,y,z) = 4x.

5.4 Triple Integrals over a Box

  1. How it worksHow does a triple integral over a rectangular box reduce to three ordinary integrals done one variable at a time?
  2. Worked exampleFind the triple integral of xyz over the box [0,1] x [0,2] x [0,3].
  3. Common mistakeWhy doesn't the order you integrate x, y, and z in matter for a triple integral over a box?
  4. Worked exampleFind the triple integral of x2 + y2 + z2 over the box [0,1] x [0,1] x [0,1].
Your turn1 of 2
Find the triple integral of 3xyz over the box [0,3] x [0,4] x [0,2].

5.4 Triple Integrals over General Regions

  1. How it worksHow do the innermost limits of a triple integral become functions of the two outer variables for a curved solid?
  2. Worked exampleFind the volume of the solid bounded above by z = 4 - x2 - y2 and below by z = 0.
  3. Going furtherWhat happens to setting up a triple integral's limits once a solid's shadow on the xy-plane isn't simple?
  4. Worked exampleFind the volume of the solid bounded by z = x + y, z = 0, x = 0, y = 0, and x + y = 1.
Your turn1 of 2
Find the volume of the solid bounded above by z = 7 - x2 - y2 and below by z = 0.

5.5 Triple Integrals in Cylindrical Coordinates

  1. How it worksHow does an extra factor of r show up in dV once a triple integral converts to cylindrical coordinates?
  2. Worked exampleFind the volume of the solid inside the cylinder x2 + y2 = 4 and between z = 0 and z = 5, using cylindrical coordinates.
  3. Worked exampleFind the mass of the solid cylinder x2 + y2 ≤ 4, 0 ≤ z ≤ 3 with density rho(x,y,z) = z, using cylindrical coordinates.
  4. Common mistakeWhy doesn't converting a triple integral to cylindrical coordinates mean the z-limits also have to be rewritten in r and theta?
Your turn1 of 2
Find the volume of the solid inside the cylinder x2 + y2 = 42 and between z = 0 and z = 7, using cylindrical coordinates.

5.5 Triple Integrals in Spherical Coordinates

  1. How it worksHow does the factor rho2 sin(phi) appear in dV once a triple integral converts to spherical coordinates?
  2. Worked exampleFind the volume of the solid sphere of radius 2, using a triple integral in spherical coordinates.
  3. Worked exampleFind the volume of the region above the cone phi = π/4 and inside the sphere rho = 2, using spherical coordinates.
  4. Common mistakeWhy isn't phi measured from the xy-plane the way latitude is, instead of from the positive z-axis?
  5. Going furtherWhat happens to the spherical volume element once phi ranges only from 0 to π/2 instead of 0 to π?
Your turn1 of 2
In converting a triple integral to spherical coordinates, write the volume element's integrand rho2 sin(phi), as a function of rho and phi.

5.5 Choosing the Right Coordinate System for a Triple Integral

  1. Ties togetherHow do the symmetries that favor cylindrical coordinates differ from the ones that favor spherical coordinates?
  2. Worked exampleFind which coordinate system best sets up the volume integral for the solid region between two concentric spheres of radius 1 and 2.
  3. Going furtherWhat happens to a triple integral's difficulty once a solid has an axis of symmetry but isn't centered at the origin?
Your turn1 of 2
Find the volume of the solid region between two concentric spheres of radius 3 and 8 (with 8 > 3).

5.7 Change of Variables and the Jacobian in Two Variables

  1. How it worksHow does the Jacobian determinant measure exactly how a small patch of area changes size under a coordinate transformation?
  2. Worked exampleFind the Jacobian of the transformation x = u + v, y = u - v.
  3. Common mistakeWhy isn't the Jacobian just the product of dx/du and dy/dv?
  4. Going furtherWhat happens to the Jacobian of a linear transformation once the transformation is just a rotation?
Your turn1 of 2
Write the Jacobian of the transformation x = au + bv, y = cu + dv, as a function of a, b, c, and d.

5.7 The Jacobian as a Local Area-Stretch Factor

  1. How it worksHow does the absolute value of the Jacobian at a point tell you the local stretching or shrinking factor there?
  2. Worked exampleFind the double integral of (x+y) ex-y over the square with vertices (0,0), (1,1), (2,0), (1,-1), using u = x+y, v = x-y.
  3. Going furtherWhat happens to a region's mapped area at a point where the Jacobian equals zero?
  4. Ties togetherHow does this local stretch factor mirror the way a cross product's magnitude measures area stretch for a parametric surface?
Your turn1 of 2
A transformation from (x,y) to (u,v) has Jacobian determinant j = partial(u,v)/partial(x,y) = 6 at a point. Find the local area-stretch factor for the REVERSE transformation from (u,v) back to (x,y), |partial(x,y)/partial(u,v)|.

5.7 Mapping an Ellipse to a Circle with a Change of Variables

  1. How it worksHow does the substitution x = au, y = bv stretch a unit circle in the uv-plane into an ellipse in the xy-plane?
  2. Worked exampleFind the double integral of 1 over the ellipse x2/4 + y2/9 ≤ 1, using the substitution x = 2u, y = 3v.
  3. Going furtherWhat happens to the Jacobian factor once the ellipse is also rotated, not just stretched along the axes?
Your turn1 of 2
Write the Jacobian of the substitution x = Au, y = Bv, as a function of A and B.

5.7 Change of Variables in Triple Integrals

  1. How it worksHow does the three-variable Jacobian generalize the area-stretch idea into a volume-stretch factor?
  2. Worked exampleShow that converting to spherical coordinates gives dV = rho2 sin(phi) d rho d phi d theta by computing the Jacobian determinant.
  3. Ties togetherHow does this general change-of-variables formula explain why cylindrical and spherical coordinates each carry their own extra factor in dV?
Your turn1 of 2
Write the Jacobian of the transformation x = au, y = bv, z = cw, as a function of a, b, and c.

5.3 Finding Area in Polar Coordinates with a Double Integral

  1. How it worksHow does integrating r dr dtheta over a polar region recover the same area a single polar integral would give?
  2. Worked exampleFind the area enclosed by one petal of r = cos(2 theta), using a double integral in polar coordinates.
  3. Worked exampleFind the area inside the circle r = 2 and outside the circle r = 1, using a double integral in polar coordinates.
Your turn1 of 2
Find the area enclosed by one petal of r = 4cos(4 theta), using a double integral in polar coordinates.