University Calculus III — Triple Integrals and Change of Variables Worksheet

All 12 questions

Each question targets one named concept from the sheet. Read them here, or print the PDF — it has working space under each one.

1. Q1Triple Integrals in Cartesian Coordinates

   Let $B$ be the rectangular box $0\le x\le 1$, $0\le y\le 2$, $0\le z\le 3$. Evaluate $${\iiint }_B(2x+yz)dV.$$

2. Q2Triple Integrals in Cartesian Coordinates

   Consider the iterated integral $${\int }_0^1{\int }_0^z{\int }_0^{y+z}6xdxdydz.$$

   1. Describe the solid of integration by a system of inequalities.
   2. Evaluate the integral.
3. Q3Setting Up the Limits for a Solid Region

   Let $E$ be the solid in the first octant ($x\ge 0$, $y\ge 0$, $z\ge 0$) lying under the plane $2x+y+2z=4$.

   1. Write ${\displaystyle {\iiint }_{E}f(x,y,z)dV}$ as an iterated integral in the order $dzdydx$.
   2. Use your limits to evaluate ${\displaystyle {\iiint }_{E}xdV}$.
4. Q4Setting Up the Limits for a Solid Region

   The solid $E$ is bounded by the parabolic cylinder $z=4-x^2$ and the planes $y=0$, $z=0$ and $y+z=4$; that is, $E$ is the set of points with $0\le z\le 4-x^2$ and $0\le y\le 4-z$.

   1. Write the volume of $E$ as an iterated integral in the order $dydzdx$.
   2. Write it again in the order $dxdzdy$.
   3. Evaluate whichever of the two is easier.
5. Q5Triple Integrals in Cylindrical Coordinates

   The solid $E$ lies inside the cylinder $x^2+y^2=9$, above the plane $z=0$ and below the cone $z=\sqrt{x^2+y^2}$. Use cylindrical coordinates $(r,\theta ,z)$ to evaluate $${\iiint }_{E}zdV.$$

6. Q6Triple Integrals in Cylindrical Coordinates

   Convert the integral $${\int }_{-2}^2{\int }_0^{\sqrt{4-x^2}}{\int }_{x^2+y^2}^{4}ydzdydx$$ to cylindrical coordinates $(r,\theta ,z)$, stating the range of each variable, and evaluate it.

7. Q7Triple Integrals in Spherical Coordinates

   Use spherical coordinates $(\rho ,\theta ,\varphi )$ in which $\varphi$ is measured from the positive $z$-axis ($0\le \varphi \le \pi$) and $\theta$ is the same angle as in cylindrical coordinates, so that $x=\rho \sin \varphi \cos \theta$, $y=\rho \sin \varphi \sin \theta$, $z=\rho \cos \varphi$ and $dV={\rho }^2\sin \varphi d\rho d\varphi d\theta$.

   Let $E$ be the part of the shell $1\le x^2+y^2+z^2\le 4$ that lies in the first octant ($x\ge 0$, $y\ge 0$, $z\ge 0$). Evaluate $${\iiint }_E(x^2+y^2+z^2)dV.$$

8. Q8Triple Integrals in Spherical Coordinates

   Use spherical coordinates $(\rho ,\theta ,\varphi )$ in which $\varphi$ is measured from the positive $z$-axis ($0\le \varphi \le \pi$) and $\theta$ is the same angle as in cylindrical coordinates, so that $x=\rho \sin \varphi \cos \theta$, $y=\rho \sin \varphi \sin \theta$, $z=\rho \cos \varphi$ and $dV={\rho }^2\sin \varphi d\rho d\varphi d\theta$.

   The solid $E$ lies inside the sphere $x^2+y^2+z^2=9$, above the plane $z=0$ and below the cone $z=\sqrt{\frac{1}{3}(x^2+y^2)}$. Find the range of each spherical variable on $E$, and evaluate $${\iiint }_{E}zdV.$$

9. Q9Volume, Mass and Centre of Mass of a Solid

   A solid cone occupies the region $\sqrt{x^2+y^2}\le z\le 2$, with lengths in metres. Its density at each point is numerically equal to the height of the point: $\delta (x,y,z)=z\text{kg}/{\text{m}}^3$. Find the mass of the cone and the coordinates of its centre of mass.

10. Q10The Jacobian and Change of Variables

    Compute the Jacobian ${\displaystyle \frac{\partial (x,y)}{\partial (u,v)}}$ of each transformation.

    1. $x=3u-v$, $y=u+2v$.
    2. $x={\displaystyle \frac{u}{v}}$, $y=uv$, for $u>0$, $v>0$. Also give its value at $(u,v)=(6,2)$.
11. Q11The Jacobian and Change of Variables

    Let $R$ be the parallelogram in the $xy$-plane bounded by the lines $x-y=0$, $x-y=2$, $x+2y=0$ and $x+2y=3$. Use the change of variables $u=x-y$, $v=x+2y$ to evaluate $${\iint }_R(x-y{)}^2(x+2y)dA.$$

12. Q12Synthesis — drawing on several topics in this unit

    The solid $E$ is enclosed between the two paraboloids $z=x^2+y^2$ and $z=8-x^2-y^2$, with lengths in centimetres.

    1. Write the volume of $E$ as an iterated integral in Cartesian coordinates, in the order $dzdydx$. Do not evaluate it.
    2. Convert to cylindrical coordinates and find the volume.
    3. The solid has density $\delta (x,y,z)=x^2+y^2\text{g}/{\text{cm}}^3$. Find its mass.
    4. State the centre of mass of the solid in part c), with a reason and without further integration.

The 9 challenge problems for this topic are a separate, paid sheet and are not reproduced here.

The rest of this unit

The worksheet is the practice, and it is free. Two more printable documents cover the same unit and come with the University Calculus III bundle: read the notes first, work this sheet, then sit the test closed-book. See what each one covers.

What the Triple Integrals and Change of Variables notes cover  About the Triple Integrals and Change of Variables unit test

How to do every concept on this sheet

This is the part worksheet sites usually leave out. Below is the actual reasoning behind each group of questions — not a full solution set, the decisions that get you to one. Read it before you start, or after you get stuck.

A triple integral over a box is three single integrals

Q1 gives a box with constant limits in x, y and z and a polynomial integrand. Exactly as over a rectangle, each integral is a partial integration: the variables not yet integrated are held constant, and each step leaves a function of the ones that remain. With constant limits any of the six orders works, so pick the one whose first step is shortest.

Q2 hands you an iterated integral and asks, before any evaluation, for the solid it describes as a system of inequalities. Read the limits from the outside in, one differential at a time, and notice which variable is innermost: it is the one whose differential is written first, and it need not be z. Its limits may contain the other two variables; the middle limits may contain only the outer variable; the outer limits are numbers.

Setting up the limits for a solid region

Over a solid that is not a box, the inner limits are surfaces: where a line parallel to the innermost axis enters the solid and where it leaves. What remains is a double integral over the solid's shadow on the plane of the other two variables, and that shadow is described exactly as a region was in the double-integral set.

Q3 is a solid in the first octant under a plane, to be written in the order dz dy dx. The top is the plane solved for z and the bottom is the floor z = 0. The shadow is a triangle in the xy-plane, and its slanted edge is the line where the plane meets that floor: set z = 0 in the plane's equation. Part (b) then uses the limits you wrote for a specific integrand.

Q4 describes a solid bounded by a parabolic cylinder and three planes, and gives its inequalities. It asks for the volume in two different orders and then leaves the choice of which to evaluate to you. The innermost variable is different in the two orders, so the shadow is on a different coordinate plane each time: draw both shadows.

Triple integrals in cylindrical coordinates

Cylindrical coordinates are polar coordinates in the xy-plane with z left alone. They pay off when the solid is built around the z-axis: a cylinder x² + y² = a² is r = a, a cone z = √(x² + y²) is z = r, a paraboloid z = x² + y² is z = r². Q5 is a solid inside a cylinder, above the xy-plane and below a cone; write each surface in r and z and the limits follow.

Q6 hands you a Cartesian iterated integral and asks you to convert it, stating the range of each variable, and evaluate. The outer two integrals describe the shadow, so read them as a region in the xy-plane first. Notice what the lower limit for y is: a shadow whose edge is a circle does not always want the full turn in θ, and the range of θ is exactly the set of directions the shadow covers. The inner limits become two surfaces written in r, and the integrand y becomes r sin θ.

Triple integrals in spherical coordinates

Spherical coordinates locate a point by its distance ρ from the origin, the angle φ measured down from the positive z-axis, and the same angle θ as in cylindrical coordinates. A sphere centred at the origin is ρ = constant, and a cone with its vertex at the origin and its axis along the z-axis is φ = constant, which is why solids bounded by those two surfaces become boxes in (ρ, θ, φ).

Q7 is a piece of a spherical shell — the region between two spheres centred at the origin — cut down to the first octant, with an integrand that is x² + y² + z². Each of the three inequalities that define the first octant fixes the range of one angle or narrows one; work out which inequality fixes which, and write the integrand in ρ before multiplying by the factor from dV.

Q8 is the question where most of the marks are in one decision. The solid is inside a sphere, above the xy-plane and below a cone, and it asks for the range of each spherical variable before the integral. Find the cone's angle by substituting the spherical formulas into its equation: ρ cancels, and what is left is an equation in φ alone.

Volume, mass and centre of mass of a solid

The formulas are the lamina's formulas with one more integral sign. The volume is the triple integral of 1; the mass is the triple integral of the density δ; and each coordinate of the centre of mass is a moment divided by the mass, where the moment about a coordinate plane carries the coordinate measured away from that plane.

Q9 is a solid cone, in metres, whose density in kilograms per cubic metre equals the height of the point. Write the cone in cylindrical coordinates first — it is one of the surfaces that coordinate system was built for. Before integrating for x̄ and ȳ, ask what the symmetry of the cone and of its density about the z-axis says about them; an integral you can settle by an argument is one you do not have to compute, but the argument has to be stated.

The Jacobian and change of variables

The r in polar and cylindrical coordinates and the ρ² sin φ in spherical ones are special cases of one idea. A map from the uv-plane to the xy-plane stretches small rectangles into small near-parallelograms, and the factor by which it multiplies area is the absolute value of a determinant of partial derivatives.

Q10 is the computation on its own, for two maps. In part (a) the map is linear, so every partial derivative is a constant and so is the Jacobian: every region has its area multiplied by the same number. In part (b) the partial derivatives depend on u and v, so the Jacobian is a function, and the question asks for its value at one point as well — which says how much a small rectangle near that point is stretched.

Q11 gives a parallelogram bounded by two pairs of parallel lines, and the substitution that goes with it. Each pair of lines is a pair of level sets of one of the new variables, so in the uv-plane the parallelogram is a rectangle with constant limits you can read straight off the four equations. The integrand is already written in terms of the two expressions the substitution names.

The synthesis question

Q12 is one solid, in centimetres, enclosed between a paraboloid opening upward and one opening downward, and it is the whole set in four parts. Part (a) is the Cartesian set-up alone, in the order dz dy dx: the bottom and top are the two paraboloids, and the shadow's edge is where they meet, which you find by setting the two heights equal. Part (b) converts that set-up to cylindrical coordinates, where both surfaces and the shadow's edge become functions of r alone, and finds the volume. Part (c) adds a density that depends only on the distance from the z-axis and asks for the mass, which is the volume integral with δ written in r inside it. Part (d) asks for the centre of mass without further integration, with a reason. That reason is a symmetry, as in Q9: name every symmetry the solid has, and for each one check that the density is unchanged by it too before you use it.

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Getting the most out of it

Name the shadow before you write a limit

Every set-up on this sheet is two surfaces over a region of a coordinate plane. Say which plane the shadow lies in, sketch it, and mark where its edge comes from — usually where the top and bottom surfaces meet. The inner limits are then only the question of which surface is below and which is above.

Translate every surface before choosing coordinates

Write each bounding surface in cylindrical and in spherical form before you commit. The system in which most of the surfaces become "variable = constant" is the one whose limits will be constants, and it is usually the right choice even when the integrand looks worse in it.

Write dV out in full every time

Write dV = dz dy dx, r dz dr dθ or ρ² sin φ dρ dφ dθ explicitly in each integral. It keeps the factor from being forgotten and keeps each pair of limits attached to the right differential.

Check a set-up by computing a volume you know

Before integrating the function the question gives you, integrate 1 with the same limits whenever the solid is a tetrahedron, a cone, a cylinder or a piece of a ball. A volume that disagrees with geometry finds a wrong limit before it costs you a whole question.

Estimate before you trust

The solid sits inside a box you can write down, so its volume is less than the box's; a centre of mass must lie inside the solid's bounding box; and a density that grows in one direction pulls the balance point that way. One line of estimate catches a missing factor or a swapped moment.

Common questions