Unit 5: Double Integrals
Volume under a bumpy surface, swept region by region, in whichever order and coordinate system makes the sweep easiest.
13 topics · 43 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.
5.1 Double Integrals over Rectangles
- How it worksHow does summing up tiny boxes of height f(x, y) over a rectangle turn into a double integral?
- Worked exampleFind the double integral of f(x, y) = xy over the rectangle 0 ≤ x ≤ 2, 0 ≤ y ≤ 3.
- Common mistakeWhy doesn't a double integral require the region underneath it to be a square just because it is called 'double'?
Your turn1 of 2
For the double integral of f(x, y) = 4·x·y over 0 ≤ y ≤ 4 (x held fixed), integrate first with respect to y. Write the result as a function of x.5.2 Iterated Integrals and Fubini's Theorem
- How it worksHow does Fubini's theorem let you evaluate a double integral as two ordinary single-variable integrals done one after the other?
- Common mistakeWhy isn't an iterated integral just two unrelated single integrals multiplied together?
- Worked exampleFind the iterated integral of x2 y over y from 0 to 2 and then x from 0 to 1, and confirm the reverse order gives the same answer.
Your turn1 of 2
For the iterated integral of f(x, y) = 4·x2·y over 0 ≤ y ≤ 4 (x held fixed), integrate first with respect to y. Write the result as a function of x.5.3 Double Integrals over General Regions
- How it worksHow do variable limits on the inner integral let a double integral sweep across a region bounded by curves instead of straight lines?
- Worked exampleFind the double integral of f(x, y) = 1 over the region bounded by y = x2 and y = x.
- Going furtherWhat happens to the limits of integration when the same region gets described with x between two functions of y instead?
Your turn1 of 2
For the region between y = 3·x2 (below) and y = 3·x (above), integrate f(x,y)=1 with respect to y from y=3·x2 to y=3·x. Write the result as a function of x.5.4 Changing the Order of Integration
- How it worksHow do you re-read a region's boundary curves to rewrite dy dx limits as dx dy limits?
- Worked exampleFind the integral of f(x, y) over 0 ≤ x ≤ 1, x ≤ y ≤ 1 by reversing the order of integration first.
- Common mistakeWhy can't you just swap the numbers sitting in the limits of integration without re-examining the region itself?
- Going furtherWhat happens to the region's shape when the inequality x ≤ y ≤ 1 is replaced by x2 ≤ y ≤ 1?
Your turn1 of 2
For the triangular region 0 ≤ x ≤ y ≤ 4, integrate f(x,y) = 4 with respect to y from y=x to y=4. Write the result as a function of x.5.5 Double Integrals in Polar Coordinates
- How it worksHow does an extra factor of r appear when dx dy converts into dr dtheta inside a polar double integral?
- Common mistakeWhy doesn't converting dx dy into dr dtheta skip the extra factor of r that area actually needs?
- Worked exampleFind the double integral of f(x, y) = x2 + y2 over the disk x2 + y2 ≤ 4 using polar coordinates.
Your turn1 of 2
Converting the double integral of 4·(x2 + y2) to polar coordinates turns the integrand into 4·r2, and dA into r dr dtheta. Write the resulting full integrand 4·r2·r, as a function of r.5.6 Area and Volume via Double Integrals
- How it worksHow does the double integral of the difference between a top surface and a bottom surface produce the volume trapped between them?
- Worked exampleFind the volume between z = 4 - x2 - y2 and the xy-plane over the disk x2 + y2 ≤ 4.
- Ties togetherHow does integrating 1 over a region to find area connect back to integrating a height function to find volume?
Your turn1 of 2
Find the volume between z = 7 - x2 - y2 and the xy-plane over the disk x2 + y2 ≤ 7.5.7 Mass and Center of Mass in the Plane
- How it worksHow does slicing a plate into infinitesimal pieces and integrating its density function build up the plate's total mass?
- Worked exampleFind the mass of a thin plate over the region 0 ≤ x ≤ 1, 0 ≤ y ≤ 1 with density rho(x, y) = x + y.
- Going furtherWhat happens to a plate's total mass when the density function is doubled everywhere at once?
Your turn1 of 2
A plate has density rho(x, y) = 4·(x + y) over 0 ≤ y ≤ 3 (x held fixed). Integrate rho with respect to y first. Write the result as a function of x.5.8 Moments and Center of Mass, General Densities
- How it worksHow do the moments about the x-axis and y-axis combine with total mass to pin down the exact center of mass?
- Common mistakeWhy doesn't the center of mass of a lopsided plate sit at the plate's plain geometric center?
- Worked exampleFind the center of mass of the plate over 0 ≤ x ≤ 2, 0 ≤ y ≤ 1 with density rho(x, y) = x.
Your turn1 of 2
Find the x-coordinate of the center of mass of the plate over 0 ≤ x ≤ 4, 0 ≤ y ≤ 6 with density rho(x, y) = 4·x.5.9 Probability Densities and Double Integrals
- How it worksHow does a joint probability density function turn a double integral over a region into a probability?
- Common mistakeWhy doesn't a joint probability density function need to equal 1 at every point, only integrate to 1 over its whole domain?
- Worked exampleFind the probability that X is between 0 and 1 and Y is between 0 and 1 for the joint density f(x, y) = xy over 0 ≤ x, y ≤ 2.
- Going furtherWhat happens to the double integral computing a joint probability when X and Y turn out to be independent random variables?
Your turn1 of 2
For the joint density f(x, y) = 3·x·y over 0 ≤ y ≤ 3 (x held fixed), integrate with respect to y first. Write the result as a function of x.5.10 Surface Area as a Double Integral
- How it worksHow do the partial derivatives fx and fy combine into the surface-area integrand square root of 1 + fx2 + fy2?
- Worked exampleFind the surface area of the part of the plane z = 2x + 2y + 1 that lies above the unit square 0 ≤ x, y ≤ 1.
- Going furtherWhat happens to the surface-area integrand when the surface being measured is perfectly flat and horizontal?
Your turn1 of 2
Find the surface area of the part of the plane z = 4·x + 4·y + 5 that lies above the unit square 0 ≤ x, y ≤ 1.5.11 Average Value of a Function over a Region
- How it worksHow does dividing the double integral of a function by the area of its region produce that function's average value?
- Worked exampleFind the average value of f(x, y) = x + y over the square 0 ≤ x ≤ 2, 0 ≤ y ≤ 2.
- Common mistakeWhy doesn't the average value of a function over a region have to equal the function's value at that region's geometric center?
- Going furtherWhat happens to a region's average value when the function being averaged is constant over that whole region?
Your turn1 of 2
For f(x, y) = 4·x + 4·y over 0 ≤ y ≤ 6 (x held fixed), integrate with respect to y first, as part of computing the average value over the square 0 ≤ x, y ≤ 6. Write the result as a function of x.5.12 Volume Between Two Surfaces Using Polar Coordinates
- How it worksHow does subtracting a bottom surface from a top surface, then converting the whole double integral to polar form, measure that solid's volume?
- Worked exampleFind the volume between the paraboloid z = x2 + y2 and the plane z = 4 using a double integral in polar coordinates.
- Going furtherWhat happens to the polar limits of integration when the region between the two surfaces is only a slice, not the full disk?
Your turn1 of 2
The volume between z = 4·(x2+y2) and z = 13, converted to polar coordinates, has integrand (13 - 4·r2)·r. Write this integrand as a function of r.5.13 Improper Double Integrals over Unbounded Regions
- How it worksHow does an improper double integral get defined as the limit of ordinary double integrals over larger and larger bounded regions?
- Worked exampleShow that the improper double integral of e−(x^2+y^2) over the entire plane converges by switching to polar coordinates.
- Common mistakeWhy doesn't a function that shrinks toward zero far from the origin automatically guarantee its improper double integral converges?
- Going furtherWhat happens to an improper double integral when the function being integrated blows up at a single interior point instead of at infinity?
Your turn1 of 2
Find the value of the improper double integral of e−4·(x^2+y^2) over the entire plane.