 6.4.1: 16 Find the volume of the solid obtained by rotating the region bou...
 6.4.2: 16 Find the volume of the solid obtained by rotating the region bou...
 6.4.3: 16 Find the volume of the solid obtained by rotating the region bou...
 6.4.4: 16 Find the volume of the solid obtained by rotating the region bou...
 6.4.5: 16 Find the volume of the solid obtained by rotating the region bou...
 6.4.6: 16 Find the volume of the solid obtained by rotating the region bou...
 6.4.7: 78 Here we rotate about the yaxis instead of the xaxis. Find the ...
 6.4.8: 78 Here we rotate about the yaxis instead of the xaxis. Find the ...
 6.4.9: Volume of a pancreas A CAT scan of a human pancreas shows crosssec...
 6.4.10: A log 10 m long is cut at 1meter intervals and its crosssectional ...
 6.4.11: (a) If the region shown in the figure is rotated about the xaxis t...
 6.4.12: Volume of a birds egg CAS (a) A model for the shape of a birds egg ...
 6.4.13: 1315 Find the volume of the described solid S. 13. S is a right cir...
 6.4.14: 1315 Find the volume of the described solid S. 14. The base of S is...
 6.4.15: 1315 Find the volume of the described solid S. 15. The base of S is...
 6.4.16: The base of S is a circular disk with radius r. Parallel crosssecti...
 6.4.17: Find the volume common to two spheres, each with radius r, if the c...
 6.4.18: Find the volume common to two circular cylinders, each with radius ...
Solutions for Chapter 6.4: Volumes
Full solutions for Biocalculus: Calculus for Life Sciences  1st Edition
ISBN: 9781133109631
Solutions for Chapter 6.4: Volumes
Get Full SolutionsChapter 6.4: Volumes includes 18 full stepbystep solutions. This textbook survival guide was created for the textbook: Biocalculus: Calculus for Life Sciences , edition: 1. This expansive textbook survival guide covers the following chapters and their solutions. Biocalculus: Calculus for Life Sciences was written by and is associated to the ISBN: 9781133109631. Since 18 problems in chapter 6.4: Volumes have been answered, more than 27416 students have viewed full stepbystep solutions from this chapter.

Associative Law (AB)C = A(BC).
Parentheses can be removed to leave ABC.

Augmented matrix [A b].
Ax = b is solvable when b is in the column space of A; then [A b] has the same rank as A. Elimination on [A b] keeps equations correct.

Back substitution.
Upper triangular systems are solved in reverse order Xn to Xl.

Cofactor Cij.
Remove row i and column j; multiply the determinant by (I)i + j •

Cross product u xv in R3:
Vector perpendicular to u and v, length Ilullllvlll sin el = area of parallelogram, u x v = "determinant" of [i j k; UI U2 U3; VI V2 V3].

Four Fundamental Subspaces C (A), N (A), C (AT), N (AT).
Use AT for complex A.

Hermitian matrix A H = AT = A.
Complex analog a j i = aU of a symmetric matrix.

Incidence matrix of a directed graph.
The m by n edgenode incidence matrix has a row for each edge (node i to node j), with entries 1 and 1 in columns i and j .

Iterative method.
A sequence of steps intended to approach the desired solution.

Krylov subspace Kj(A, b).
The subspace spanned by b, Ab, ... , AjIb. Numerical methods approximate A I b by x j with residual b  Ax j in this subspace. A good basis for K j requires only multiplication by A at each step.

Left inverse A+.
If A has full column rank n, then A+ = (AT A)I AT has A+ A = In.

Linear transformation T.
Each vector V in the input space transforms to T (v) in the output space, and linearity requires T(cv + dw) = c T(v) + d T(w). Examples: Matrix multiplication A v, differentiation and integration in function space.

Network.
A directed graph that has constants Cl, ... , Cm associated with the edges.

Pascal matrix
Ps = pascal(n) = the symmetric matrix with binomial entries (i1~;2). Ps = PL Pu all contain Pascal's triangle with det = 1 (see Pascal in the index).

Rank r (A)
= number of pivots = dimension of column space = dimension of row space.

Rayleigh quotient q (x) = X T Ax I x T x for symmetric A: Amin < q (x) < Amax.
Those extremes are reached at the eigenvectors x for Amin(A) and Amax(A).

Row picture of Ax = b.
Each equation gives a plane in Rn; the planes intersect at x.

Semidefinite matrix A.
(Positive) semidefinite: all x T Ax > 0, all A > 0; A = any RT R.

Symmetric factorizations A = LDLT and A = QAQT.
Signs in A = signs in D.

Vector v in Rn.
Sequence of n real numbers v = (VI, ... , Vn) = point in Rn.