- 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 y-axis instead of the x-axis. Find the ...
- 6.4.8: 78 Here we rotate about the y-axis instead of the x-axis. Find the ...
- 6.4.9: Volume of a pancreas A CAT scan of a human pancreas shows cross-sec...
- 6.4.10: A log 10 m long is cut at 1-meter intervals and its crosssectional ...
- 6.4.11: (a) If the region shown in the figure is rotated about the x-axis 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
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.
Upper triangular systems are solved in reverse order Xn to Xl.
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 edge-node incidence matrix has a row for each edge (node i to node j), with entries -1 and 1 in columns i and j .
A sequence of steps intended to approach the desired solution.
Krylov subspace Kj(A, b).
The subspace spanned by b, Ab, ... , Aj-Ib. 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.
A directed graph that has constants Cl, ... , Cm associated with the edges.
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.