 11.5.11.5.1: Fill in the blanks. ni1i
 11.5.11.5.2: Fill in the blanks. ni1i3
 11.5.11.5.3: Fill in the blanks. The exact _______ of a plane region is given by...
 11.5.11.5.4: In Exercises 18, evaluate the sum using the summation formulas and ...
 11.5.11.5.5: In Exercises 18, evaluate the sum using the summation formulas and ...
 11.5.11.5.6: In Exercises 18, evaluate the sum using the summation formulas and ...
 11.5.11.5.7: In Exercises 18, evaluate the sum using the summation formulas and ...
 11.5.11.5.8: In Exercises 18, evaluate the sum using the summation formulas and ...
 11.5.11.5.9: In Exercises 916, (a) rewrite the sum as a rational function (b) Us...
 11.5.11.5.10: In Exercises 916, (a) rewrite the sum as a rational function (b) Us...
 11.5.11.5.11: In Exercises 916, (a) rewrite the sum as a rational function (b) Us...
 11.5.11.5.12: In Exercises 916, (a) rewrite the sum as a rational function (b) Us...
 11.5.11.5.13: In Exercises 916, (a) rewrite the sum as a rational function (b) Us...
 11.5.11.5.14: In Exercises 916, (a) rewrite the sum as a rational function (b) Us...
 11.5.11.5.15: In Exercises 916, (a) rewrite the sum as a rational function (b) Us...
 11.5.11.5.16: In Exercises 916, (a) rewrite the sum as a rational function (b) Us...
 11.5.11.5.17: In Exercises 1720, approximate the area of the region using the ind...
 11.5.11.5.18: In Exercises 1720, approximate the area of the region using the ind...
 11.5.11.5.19: In Exercises 1720, approximate the area of the region using the ind...
 11.5.11.5.20: In Exercises 1720, approximate the area of the region using the ind...
 11.5.11.5.21: In Exercises 2124, complete the table showing the approximate area ...
 11.5.11.5.22: In Exercises 2124, complete the table showing the approximate area ...
 11.5.11.5.23: In Exercises 2124, complete the table showing the approximate area ...
 11.5.11.5.24: In Exercises 2124, complete the table showing the approximate area ...
 11.5.11.5.25: In Exercises 2532, complete the table using the function over the s...
 11.5.11.5.26: In Exercises 2532, complete the table using the function over the s...
 11.5.11.5.27: In Exercises 2532, complete the table using the function over the s...
 11.5.11.5.28: In Exercises 2532, complete the table using the function over the s...
 11.5.11.5.29: In Exercises 2532, complete the table using the function over the s...
 11.5.11.5.30: In Exercises 2532, complete the table using the function over the s...
 11.5.11.5.31: In Exercises 2532, complete the table using the function over the s...
 11.5.11.5.32: In Exercises 2532, complete the table using the function over the s...
 11.5.11.5.33: In Exercises 33 44, use the limit process to find the area of the r...
 11.5.11.5.34: In Exercises 33 44, use the limit process to find the area of the r...
 11.5.11.5.35: In Exercises 33 44, use the limit process to find the area of the r...
 11.5.11.5.36: In Exercises 33 44, use the limit process to find the area of the r...
 11.5.11.5.37: In Exercises 33 44, use the limit process to find the area of the r...
 11.5.11.5.38: In Exercises 33 44, use the limit process to find the area of the r...
 11.5.11.5.39: In Exercises 33 44, use the limit process to find the area of the r...
 11.5.11.5.40: In Exercises 33 44, use the limit process to find the area of the r...
 11.5.11.5.41: In Exercises 33 44, use the limit process to find the area of the r...
 11.5.11.5.42: In Exercises 33 44, use the limit process to find the area of the r...
 11.5.11.5.43: In Exercises 33 44, use the limit process to find the area of the r...
 11.5.11.5.44: In Exercises 33 44, use the limit process to find the area of the r...
 11.5.11.5.45: Geometry The boundaries of a parcel of land are two edges modeled b...
 11.5.11.5.46: Geometry The table shows the measurements (in feet) of a lot bounde...
 11.5.11.5.47: True or False? In Exercises 47 and 48, determine whether the statem...
 11.5.11.5.48: True or False? In Exercises 47 and 48, determine whether the statem...
 11.5.11.5.49: Writing Describe the process of finding the area of a region bounde...
 11.5.11.5.50: Think About It Determine which value best approximates the area of ...
 11.5.11.5.51: In Exercises 51 and 52, solve the equation. 2 tan x tan 2x c
 11.5.11.5.52: In Exercises 51 and 52, solve the equation. cos 2x 3 sin x 23
 11.5.11.5.53: In Exercises 5356, use the vectors and to find the indicated quantity
 11.5.11.5.54: In Exercises 5356, use the vectors and to find the indicated quantity
 11.5.11.5.55: In Exercises 5356, use the vectors and to find the indicated quantity
 11.5.11.5.56: In Exercises 5356, use the vectors and to find the indicated quantity
Solutions for Chapter 11.5: The Area Problem
Full solutions for Precalculus With Limits A Graphing Approach  5th Edition
ISBN: 9780618851522
Solutions for Chapter 11.5: The Area Problem
Get Full SolutionsSince 56 problems in chapter 11.5: The Area Problem have been answered, more than 102154 students have viewed full stepbystep solutions from this chapter. This expansive textbook survival guide covers the following chapters and their solutions. Chapter 11.5: The Area Problem includes 56 full stepbystep solutions. This textbook survival guide was created for the textbook: Precalculus With Limits A Graphing Approach, edition: 5. Precalculus With Limits A Graphing Approach was written by and is associated to the ISBN: 9780618851522.

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.

Big formula for n by n determinants.
Det(A) is a sum of n! terms. For each term: Multiply one entry from each row and column of A: rows in order 1, ... , nand column order given by a permutation P. Each of the n! P 's has a + or  sign.

Cramer's Rule for Ax = b.
B j has b replacing column j of A; x j = det B j I det A

Dimension of vector space
dim(V) = number of vectors in any basis for V.

Distributive Law
A(B + C) = AB + AC. Add then multiply, or mUltiply then add.

Echelon matrix U.
The first nonzero entry (the pivot) in each row comes in a later column than the pivot in the previous row. All zero rows come last.

Elimination matrix = Elementary matrix Eij.
The identity matrix with an extra eij in the i, j entry (i # j). Then Eij A subtracts eij times row j of A from row i.

Full row rank r = m.
Independent rows, at least one solution to Ax = b, column space is all of Rm. Full rank means full column rank or full row rank.

Fundamental Theorem.
The nullspace N (A) and row space C (AT) are orthogonal complements in Rn(perpendicular from Ax = 0 with dimensions rand n  r). Applied to AT, the column space C(A) is the orthogonal complement of N(AT) in Rm.

Matrix multiplication AB.
The i, j entry of AB is (row i of A)ยท(column j of B) = L aikbkj. By columns: Column j of AB = A times column j of B. By rows: row i of A multiplies B. Columns times rows: AB = sum of (column k)(row k). All these equivalent definitions come from the rule that A B times x equals A times B x .

Orthogonal matrix Q.
Square matrix with orthonormal columns, so QT = Ql. Preserves length and angles, IIQxll = IIxll and (QX)T(Qy) = xTy. AlllAI = 1, with orthogonal eigenvectors. Examples: Rotation, reflection, permutation.

Reduced row echelon form R = rref(A).
Pivots = 1; zeros above and below pivots; the r nonzero rows of R give a basis for the row space of A.

Schur complement S, D  C A } B.
Appears in block elimination on [~ g ].

Solvable system Ax = b.
The right side b is in the column space of A.

Spanning set.
Combinations of VI, ... ,Vm fill the space. The columns of A span C (A)!

Spectral Theorem A = QAQT.
Real symmetric A has real A'S and orthonormal q's.

Toeplitz matrix.
Constant down each diagonal = timeinvariant (shiftinvariant) filter.

Vandermonde matrix V.
V c = b gives coefficients of p(x) = Co + ... + Cn_IXn 1 with P(Xi) = bi. Vij = (Xi)jI and det V = product of (Xk  Xi) for k > i.

Vector addition.
v + w = (VI + WI, ... , Vn + Wn ) = diagonal of parallelogram.

Volume of box.
The rows (or the columns) of A generate a box with volume I det(A) I.