 15.1: Calculate the Riemann sum S2,3 for 4 1 6 2 x2y dx dy using two choi...
 15.2: Let SN ,N be the Riemann sum for 1 0 1 0 cos(xy) dx dy using midpoi...
 15.3: Let D be the shaded domain in Figure 1. D x 0.5 1 1.5 2 0.5 1 1.5 2...
 15.4: Explain the following: (a) 1 1 1 1 sin(xy) dx dy = 0 (b) 1 1 1 1 co...
 15.5: In Exercises 58, evaluate the iterated integral. 5. 2 0 5 3 y(x y) ...
 15.6: In Exercises 58, evaluate the iterated integral.0 1/2 /6 0 e2y sin(...
 15.7: In Exercises 58, evaluate the iterated integral./3 0 /6 0 sin(x + y...
 15.8: In Exercises 58, evaluate the iterated integral.2 1 2 1 y dx dy x +...
 15.9: In Exercises 914, sketch the domainD and calculate D f (x, y) dA.D ...
 15.10: In Exercises 914, sketch the domainD and calculate D f (x, y) dA. D...
 15.11: In Exercises 914, sketch the domainD and calculate D f (x, y) dA. D...
 15.12: In Exercises 914, sketch the domainD and calculate D f (x, y) dA. D...
 15.13: In Exercises 914, sketch the domainD and calculate D f (x, y) dA. D...
 15.14: In Exercises 914, sketch the domainD and calculate D f (x, y) dA. D...
 15.15: Express 3 3 9x2 0 f (x, y) dy dx as an iterated integral in the ord...
 15.16: Let W be the region bounded by the planes y = z, 2y + z = 3, and z ...
 15.17: Let D be the domain between y = x and y = x. Calculate D xy d
 15.18: Find the double integral of f (x, y) = x3y over the region between ...
 15.19: Change the order of integration and evaluate 9 0 y 0 x dx dy (x2 + ...
 15.20: Verify directly that 3 2 2 0 dy dx 1 + x y = 2 0 3 2 dx dy 1 + x y 21.
 15.21: Prove the formula 1 0 y 0 f (x) dx dy = 1 0 (1 x)f (x) dx Then use ...
 15.22: Rewrite 1 0 1y2 1y2 y dx dy (1 + x2 + y2) 2 by interchanging the or...
 15.23: Use cylindrical coordinates to compute the volume of the region def...
 15.24: Evaluate D x dA, where D is the shaded domain in Figure 2. 2
 15.25: Find the volume of the region between the graph of the function f (...
 15.26: Evaluate 3 0 4 1 4 2 (x3 + y2 + z) dx dy dz. 2
 15.27: Calculate B (xy + z) dV , where B = 0 x 2, 0 y 1, 1 z 3 as an itera...
 15.28: Calculate W xyz dV , where W = 0 x 1, x y 1, x z x + y 29
 15.29: Evaluate I = 1 1 1x2 0 1 0 (x + y + z) dz dy dx. 30
 15.30: Describe a region whose volume is equal to: (a) 2 0 /2 0 9 4 2 sin ...
 15.31: Find the volume of the solid contained in the cylinder x2 + y2 = 1 ...
 15.32: Use polar coordinates to evaluate D x dA, whereD is the shaded regi...
 15.33: Use polar coordinates to calculate D x2 + y2 dA, where D is the reg...
 15.34: Calculate D sin(x2 + y2)dA, where D = % 2 x2 + y2 & 3
 15.35: Express in cylindrical coordinates and evaluate: 1 0 1x2 0 x2+y2 0 ...
 15.36: Use spherical coordinates to calculate the triple integral of f (x,...
 15.37: Convert to spherical coordinates and evaluate: 2 2 4x2 4x2 4x2y2 0 ...
 15.38: Find the average value of f (x, y, z) = xy2z3 on the box [0, 1] [0,...
 15.39: Let W be the ball of radius R in R3 centered at the origin, and let...
 15.40: Express the average value of f (x, y) = exy over the ellipse x2 2 +...
 15.41: Use cylindrical coordinates to find the mass of the solid bounded b...
 15.42: Let W be the portion of the halfcylinder x2 + y2 4, x 0 such that ...
 15.43: Use cylindrical coordinates to find the mass of a cylinder of radiu...
 15.44: Find the centroid of the region W bounded, in spherical coordinates...
 15.45: Find the centroid of the solid bounded by the xyplane, the cylinde...
 15.46: Using cylindrical coordinates, prove that the centroid of a right c...
 15.47: Find the centroid of solid (A) in Figure 4 defined by x2 + y2 R2, 0...
 15.48: Calculate the coordinate yCM of the centroid of solid (B) in Figure...
 15.49: Find the center of mass of the cylinder x2 + y2 = 1 for 0 z 1, assu...
 15.50: Find the center of mass of the sector of central angle 20 (symmetri...
 15.51: Find the center of mass of the first octant of the ball x2 + y2 + z...
 15.52: Find a constant C such that p(x, y) = " C(4x y + 3) if 0 x 2 and 0 ...
 15.53: Calculate P (3X + 2Y 6) for the probability density in Exercise 52.
 15.54: The lifetimes X and Y (in years) of two machine components have joi...
 15.55: An insurance company issues two kinds of policies: A and B. Let X b...
 15.56: Compute the Jacobian of the map G(r, s) = er cosh(s), er sinh(s)
 15.57: Find a linear mapping G(u, v) that maps the unit square to the para...
 15.58: Use the map G(u, v) = u + v 2 , u v 2 to compute R (x y)sin(x + y) ...
 15.59: Let D be the shaded region in Figure 6, and let F be the map u = y ...
 15.60: Calculate the integral of f (x, y) = e3x2y over the parallelogram i...
 15.61: Sketch the region D bounded by the curves y = 2/x, y = 1/(2x), y = ...
Solutions for Chapter 15: MULTIPLE INTEGRATION
Full solutions for Calculus: Early Transcendentals  3rd Edition
ISBN: 9781464114885
Solutions for Chapter 15: MULTIPLE INTEGRATION
Get Full SolutionsThis textbook survival guide was created for the textbook: Calculus: Early Transcendentals , edition: 3. This expansive textbook survival guide covers the following chapters and their solutions. Chapter 15: MULTIPLE INTEGRATION includes 61 full stepbystep solutions. Calculus: Early Transcendentals was written by and is associated to the ISBN: 9781464114885. Since 61 problems in chapter 15: MULTIPLE INTEGRATION have been answered, more than 41868 students have viewed full stepbystep solutions from this chapter.

Absolute value of a vector
See Magnitude of a vector.

Algebraic expression
A combination of variables and constants involving addition, subtraction, multiplication, division, powers, and roots

Anchor
See Mathematical induction.

Angle between vectors
The angle formed by two nonzero vectors sharing a common initial point

Average rate of change of ƒ over [a, b]
The number ƒ(b)  ƒ(a) b  a, provided a ? b.

Coordinate(s) of a point
The number associated with a point on a number line, or the ordered pair associated with a point in the Cartesian coordinate plane, or the ordered triple associated with a point in the Cartesian threedimensional space

Factoring (a polynomial)
Writing a polynomial as a product of two or more polynomial factors.

Halflife
The amount of time required for half of a radioactive substance to decay.

Initial point
See Arrow.

Integrable over [a, b] Lba
ƒ1x2 dx exists.

Irreducible quadratic over the reals
A quadratic polynomial with real coefficients that cannot be factored using real coefficients.

Minor axis
The perpendicular bisector of the major axis of an ellipse with endpoints on the ellipse.

Parametric equations for a line in space
The line through P0(x 0, y0, z 0) in the direction of the nonzero vector v = <a, b, c> has parametric equations x = x 0 + at, y = y 0 + bt, z = z0 + ct.

Quadratic formula
The formula x = b 2b2  4ac2a used to solve ax 2 + bx + c = 0.

Reflection across the xaxis
x, y and (x,y) are reflections of each other across the xaxis.

Sample survey
A process for gathering data from a subset of a population, usually through direct questioning.

Semiperimeter of a triangle
Onehalf of the sum of the lengths of the sides of a triangle.

Sum of a finite arithmetic series
Sn = na a1 + a2 2 b = n 2 32a1 + 1n  12d4,

Sum of a finite geometric series
Sn = a111  r n 2 1  r

Variable (in statistics)
A characteristic of individuals that is being identified or measured.