 12.1: A contour map is shown for a function on the square . Use a Riemann...
 12.2: Use the Midpoint Rule to estimate the integral in Exercise 1
 12.3: Calculate the iterated integral.
 12.4: Calculate the iterated integral.
 12.5: Calculate the iterated integral.
 12.6: Calculate the iterated integral.
 12.7: Calculate the iterated integral.
 12.8: Calculate the iterated integral.
 12.9: Write as an iterated integral, where is the region shown and is an ...
 12.10: Write as an iterated integral, where is the region shown and is an ...
 12.11: Describe the region whose area is given by the integral
 12.12: Describe the solid whose volume is given by the integral and evalua...
 12.13: Calculate the iterated integral by first reversing the order of int...
 12.14: Calculate the iterated integral by first reversing the order of int...
 12.15: Calculate the value of the multiple integral.
 12.16: Calculate the value of the multiple integral.
 12.17: Calculate the value of the multiple integral.
 12.18: Calculate the value of the multiple integral.
 12.19: Calculate the value of the multiple integral.
 12.20: Calculate the value of the multiple integral.
 12.21: Calculate the value of the multiple integral.
 12.22: Calculate the value of the multiple integral.
 12.23: Calculate the value of the multiple integral.
 12.24: Calculate the value of the multiple integral.
 12.25: Calculate the value of the multiple integral.
 12.26: Calculate the value of the multiple integral.
 12.27: Calculate the value of the multiple integral.
 12.28: Calculate the value of the multiple integral.
 12.29: Find the volume of the given solid.Under the paraboloid and above t...
 12.30: Find the volume of the given solid.. Under the surface and above th...
 12.31: Find the volume of the given solid.The solid tetrahedron with vertices
 12.32: Find the volume of the given solid.Bounded by the cylinder and the ...
 12.33: Find the volume of the given solid.One of the wedges cut from the c...
 12.34: Find the volume of the given solid.Above the paraboloid and below t...
 12.35: Consider a lamina that occupies the region bounded by the parabola ...
 12.36: A lamina occupies the part of the disk that lies in the first quadr...
 12.37: (a) Find the centroid of a right circular cone with height and base...
 12.38: Find the center of mass of the solid tetrahedron with vertices , , ...
 12.39: The cylindrical coordinates of a point are . Find the rectangular a...
 12.40: The rectangular coordinates of a point are . Find the cylindrical a...
 12.41: The spherical coordinates of a point are . Find the rectangular and...
 12.42: . Identify the surfaces whose equations are given.
 12.43: Write the equation in cylindrical coordinates and in spherical coor...
 12.44: Sketch the solid consisting of all points with spherical coordinate...
 12.45: Use polar coordinates to evaluate
 12.46: Use spherical coordinates to evaluate
 12.47: Rewrite the integral as an iterated integral in the orde
 12.48: Give five other iterated integrals that are equal to
 12.49: Use the transformation , to evaluate , where is the square with ver...
 12.50: Use the transformation , , to find the volume of the region bounded...
 12.51: Use the change of variables formula and an appropriate transformati...
 12.52: (a) Evaluate , where is an integer and is the region bounded by the...
Solutions for Chapter 12: Multiple Integrals
Full solutions for Essential Calculus  2nd Edition
ISBN: 9781133112297
Solutions for Chapter 12: Multiple Integrals
Get Full SolutionsSummary of Chapter 12: Multiple Integrals
In this chapter we extend the idea of a definite integral to double and triple integrals of functions of two or three variables.
Chapter 12: Multiple Integrals includes 52 full stepbystep solutions. Essential Calculus was written by and is associated to the ISBN: 9781133112297. This expansive textbook survival guide covers the following chapters and their solutions. This textbook survival guide was created for the textbook: Essential Calculus, edition: 2. Since 52 problems in chapter 12: Multiple Integrals have been answered, more than 26509 students have viewed full stepbystep solutions from this chapter.

Causation
A relationship between two variables in which the values of the response variable are directly affected by the values of the explanatory variable

Conditional probability
The probability of an event A given that an event B has already occurred

Cotangent
The function y = cot x

Doubleblind experiment
A blind experiment in which the researcher gathering data from the subjects is not told which subjects have received which treatment

End behavior asymptote of a rational function
A polynomial that the function approaches as.

Equivalent arrows
Arrows that have the same magnitude and direction.

Exponential function
A function of the form ƒ(x) = a ? bx,where ?0, b > 0 b ?1

Graph of a relation
The set of all points in the coordinate plane corresponding to the ordered pairs of the relation.

Graph of parametric equations
The set of all points in the coordinate plane corresponding to the ordered pairs determined by the parametric equations.

Index of summation
See Summation notation.

Inverse cosine function
The function y = cos1 x

Law of sines
sin A a = sin B b = sin C c

Linear combination of vectors u and v
An expression au + bv , where a and b are real numbers

Mapping
A function viewed as a mapping of the elements of the domain onto the elements of the range

Monomial function
A polynomial with exactly one term.

Numerical derivative of ƒ at a
NDER f(a) = ƒ1a + 0.0012  ƒ1a  0.00120.002

Parametric curve
The graph of parametric equations.

Quartic function
A degree 4 polynomial function.

Scalar
A real number.

Sum identity
An identity involving a trigonometric function of u + v