High School
SAT
ACT
University
MCAT
LSAT
DAT
Wize University Calculus 2 Textbook > Bonus: Integral calculus in several variables (Videos Coming Soon)

Practice quiz: Multiple integrals

Applications of multiple integralsPrevious Section
Popular Courses
Find My Course
Multiple Integrals Quiz
Compute ∬Dx2ey   ⁣dA\displaystyle\iint_D x^2e^y \ \de{A}, where DD is the region −1≤x≤1−1 \leq x \leq 1 and 0≤y≤10 \leq y \leq 1.
Write down an iterated integral representing ∬Df(x,y)dA\displaystyle\iint_Df(x,y)dA where DD is the region bounded by x=y2−4x = y^2 − 4 and x=4−y2x = 4− y^2.
Write down a polar integral representing the volume under the surface z=sin⁡(x2+y2)z=\sin(x^2+y^2) where 1≤x2+y2≤21\le x^2+y^2\le2 and y≤0y\le0.
Compute ∬D(x2+y2)−1/2dA\displaystyle\iint \limits_D (x^2 + y^2)^{−1/2}dA, where DD is the region bounded by y=∣x∣y = |x|, x2+y2≤16x^2 + y^2 \leq 16, and y≥0y \geq 0.
Evaluate the double integral I=∬Dsin⁡(x2+y2)   ⁣dA\displaystyle I=\iint_D^{ }\sin\left(x^2+y^2\right)\ \de{A}, where DD is the region inside a circle of radius 3 on the first quadrant.

Find the area of the region which lies outside the circle x2+y2=1x^2+y^2=1 but inside the circle x2+(y−1)2=1x^2+(y−1)^2=1.
Compute ∫01∫02∫12(xy+yz)dxdydz\displaystyle\int^1_0\int^2_0\int^2_1(xy + yz) dxdydz.
Compute ∭EzdV\displaystyle\iiint_EzdV, where EE is the region bounded by the planes x=0x = 0, y=0y = 0, z=0z = 0, x+z=1x + z = 1, and 2y+z=12y + z = 1.
Compute ∭ExdV\displaystyle\iiint_ExdV, where EE is the region under the plane x+2y+3z=6x + 2y + 3z = 6 that lies in the octant x,y,z≥0x, y, z \geq 0.
Write down an integral using cylindrical coordinates representing ∭Eln⁡(x2+y2+z2)   ⁣dV\displaystyle\iiint_E\ln(x^2 + y^2 + z^2) \ \de{V} where EE is the region inside the cylinder x2+y2=4x^2 + y^2 = 4, outside the cylinder x2+y2=1x^2 + y^2 = 1, and inside the sphere x2+y2+z2=9x^2 + y^2 + z^2 = 9.
Write down an integral using spherical coordinates representing ∭E(x2+y2)   ⁣dV\displaystyle\iiint_E(x^2+y^2) \ \de{V} where EE is the part of the positive octant inside the sphere x2+y2+z2=4x^2+y^2+z^2 = 4, outside the sphere x2+y2+z2=2x^2+y^2+z^2 = 2, and inside the cone 2z=x2+y22z =\sqrt{x^2 + y^2}.
Convert the following integral into cylindrical coordinates: ∫−11∫01−x2∫x2+y2x2+y2xy   ⁣dz  ⁣dy  ⁣dx\displaystyle \int^1_{−1}\int^{\sqrt{1−x^2}}_0\int^{\sqrt{x^2+y^2}}_{x^2+y^2}xy \ \de{z}\de{y}\de{x}.

Convert the following integral into spherical coordinates: ∫04∫04−x2∫04−x2−y2x2+y2+z21+x2+y2+z2   ⁣dz  ⁣dy  ⁣dx\displaystyle\int^4_0\int^{\sqrt{4−x^2}}_0\int^{\sqrt{4−x^2−y^2}}_0\frac{\sqrt{x^2 + y^2 + z^2}}{1 + x^2 + y^2 + z^2} \ \de{z}\de{y}\de{x}.

Let EE be the solid bounded by z2=x2+y2z^2=x^2+y^2, x2+y2+z2=1x^2+y^2+z^2=1, and z≥0z\ge0. Evaluate ∭Ez   ⁣dV\displaystyle\iiint \limits_E z \ \de{V}.
.
Evaluate ∭Eex2+y2+z2x2+y2   ⁣dV\displaystyle\iiint_E\frac{e^{x^2+y^2+z^2}}{\sqrt{x^2 + y^2}} \ \de{V}, where EE is the solid bounded by the spheres of radii 1 and 2 centered at the origin and outside the double cone z2=x2+y2z^2 = x^2 + y^2.
Centroid, centre of mass, moment of inertia Quiz
Find the centre of mass of a cube bounded by the planes 0≤x,y,z≤20 \le x, y, z\le2 with density δ(x,y,z)=x2+y2+z2.\delta(x, y, z) = x^2 + y^2 + z^2.
Find the mass and center of mass of the solid region bounded by the plane z=4z=4 and z=x2+y2z=x^2+y^2. The density function is ρ(x,y,z)=3+x\rho\left(x,y,z\right)=3+x.

Find the centroid of a hemisphere of radius RR.
Find the center of mass of a semicircular plate with a constant density of BBand a radius of aa.
A flat plate is defined by the inequalities x2+y2≤4x^2+y^2\le4, 0≤y≤10\le y\le1, x≤0x\le0. The density of the plate is −xy-xy. Find the mass of the plate.
A flat plate is in the shape of the region RR in the first quadrant lying between x2+y2=1x^2+y^2=1 and x2+y2=4x^2+y^2=4. The density of the plate is x+yx+y per unit area. Find the mass of the area.
checklist
Mark Yourself Question
  1. Grab a piece of paper and try this problem yourself.
  2. When you're done, check the "I have answered this question" box below.
  3. View the solution and report whether you got it right or wrong.
Find the x-coordinate of the centroid of the region bounded by y = sin x, y =cos x, x =0, and x=π/4.x = \pi/4.
Consider the tetrahedron bounded by x + y + z = 1 in the positive octant. Find the moment of inertia of this object about the z-axis if it has a constant density of 2.