 5.2.1: Evaluate the Riemann sum for fsxd 3 2 1 2 x, 2 < x < 14, with six s...
 5.2.2: If fsxd x 2 2 2x, 0 < x < 3, evaluate the Riemann sum with n 6, tak...
 5.2.3: If fsxd ex 2 2, 0 < x < 2, find the Riemann sum with n 4 correct to...
 5.2.4: (a) Find the Riemann sum for fsxd sin x, 0 < x < 3y2, with six term...
 5.2.5: The graph of a function f is given. Estimate y 8 0 fsxd dx using fo...
 5.2.6: The graph of t is shown. Estimate y 3 23 tsxd dx with six subinterv...
 5.2.7: A table of values of an increasing function f is shown. Use the tab...
 5.2.8: The table gives the values of a function obtained from an experimen...
 5.2.9: 912 Use the Midpoint Rule with the given value of n to approximate ...
 5.2.10: 912 Use the Midpoint Rule with the given value of n to approximate ...
 5.2.11: 912 Use the Midpoint Rule with the given value of n to approximate ...
 5.2.12: 912 Use the Midpoint Rule with the given value of n to approximate ...
 5.2.13: Drug pharmacokinetics During testing of a new drug, researchers mea...
 5.2.14: Salicylic acid pharmacokinetics In the study cited in Example 5, th...
 5.2.15: 1518 Express the limit as a definite integral on the given interval...
 5.2.16: 1518 Express the limit as a definite integral on the given interval...
 5.2.17: 1518 Express the limit as a definite integral on the given interval...
 5.2.18: 1518 Express the limit as a definite integral on the given interval...
 5.2.19: 1923 Use the form of the definition of the integral given in Theore...
 5.2.20: 1923 Use the form of the definition of the integral given in Theore...
 5.2.21: 1923 Use the form of the definition of the integral given in Theore...
 5.2.22: 1923 Use the form of the definition of the integral given in Theore...
 5.2.23: 1923 Use the form of the definition of the integral given in Theore...
 5.2.24: (a) Find an approximation to the integral y 4 0 sx 2 2 3xd dx using...
 5.2.25: 2526 Express the integral as a limit of Riemann sums. Do not evalua...
 5.2.26: 2526 Express the integral as a limit of Riemann sums. Do not evalua...
 5.2.27: The graph of f is shown. Evaluate each integral by interpreting it ...
 5.2.28: The graph of t consists of two straight lines and a semicircle. Use...
 5.2.29: 2934 Evaluate the integral by interpreting it in terms of areas. 29...
 5.2.30: 2934 Evaluate the integral by interpreting it in terms of areas. 29...
 5.2.31: 2934 Evaluate the integral by interpreting it in terms of areas. 29...
 5.2.32: 2934 Evaluate the integral by interpreting it in terms of areas. 29...
 5.2.33: 2934 Evaluate the integral by interpreting it in terms of areas. 29...
 5.2.34: 2934 Evaluate the integral by interpreting it in terms of areas. 29...
 5.2.35: Evaluate y sin2 x cos4 x dx.
 5.2.36: Given that y 1 0 3xsx 2 1 4 dx 5s5 2 8, what is y 0 1 3usu2 1 4 du?
 5.2.37: Write as a single integral in the form y b a fsxd dx: y 2 22 fsxd d...
 5.2.38: If y 5 1 fsxd dx 12 and y 5 4 fsxd dx 3.6, find y 4 1 fsxd dx.
 5.2.39: If y 9 0 fsxd dx 37 and y 9 0 tsxd dx 16, find y 9 0 f2 fsxd 1 3tsx...
 5.2.40: Find y 5 0 fsxd dx if fsxd H 3 for x , 3 x for x > 3
 5.2.41: For the function f whose graph is shown, list the following quantit...
 5.2.42: If Fsxd y x 2 fstd dt, where f is the function whose graph is given...
 5.2.43: Each of the regions A, B, and C bounded by the graph of f and the x...
 5.2.44: Suppose f has absolute minimum value m and absolute maximum value M...
 5.2.45: Use the properties of integrals to verify that 2 < y 1 21 s1 1 x 2 ...
 5.2.46: Use Property 8 to estimate the value of the integral y 2 0 1 1 1 x ...
 5.2.47: 4748 Express the limit as a definite integral. 47. lim n l` o n i1 ...
 5.2.48: 4748 Express the limit as a definite integral. 47. lim n l` o n i1 ...
Solutions for Chapter 5.2: The Definite Integral
Full solutions for Biocalculus: Calculus for Life Sciences  1st Edition
ISBN: 9781133109631
Solutions for Chapter 5.2: The Definite Integral
Get Full SolutionsBiocalculus: Calculus for Life Sciences was written by and is associated to the ISBN: 9781133109631. This textbook survival guide was created for the textbook: Biocalculus: Calculus for Life Sciences , edition: 1. Since 48 problems in chapter 5.2: The Definite Integral have been answered, more than 24828 students have viewed full stepbystep solutions from this chapter. Chapter 5.2: The Definite Integral includes 48 full stepbystep solutions. This expansive textbook survival guide covers the following chapters and their solutions.

Basis for V.
Independent vectors VI, ... , v d whose linear combinations give each vector in V as v = CIVI + ... + CdVd. V has many bases, each basis gives unique c's. A vector space has many bases!

Block matrix.
A matrix can be partitioned into matrix blocks, by cuts between rows and/or between columns. Block multiplication ofAB is allowed if the block shapes permit.

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].

Dot product = Inner product x T y = XI Y 1 + ... + Xn Yn.
Complex dot product is x T Y . Perpendicular vectors have x T y = O. (AB)ij = (row i of A)T(column j of B).

Eigenvalue A and eigenvector x.
Ax = AX with x#O so det(A  AI) = o.

Factorization
A = L U. If elimination takes A to U without row exchanges, then the lower triangular L with multipliers eij (and eii = 1) brings U back to A.

GaussJordan method.
Invert A by row operations on [A I] to reach [I AI].

Hankel matrix H.
Constant along each antidiagonal; hij depends on i + j.

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

Length II x II.
Square root of x T x (Pythagoras in n dimensions).

Nilpotent matrix N.
Some power of N is the zero matrix, N k = o. The only eigenvalue is A = 0 (repeated n times). Examples: triangular matrices with zero diagonal.

Norm
IIA II. The ".e 2 norm" of A is the maximum ratio II Ax II/l1x II = O"max· Then II Ax II < IIAllllxll and IIABII < IIAIIIIBII and IIA + BII < IIAII + IIBII. Frobenius norm IIAII} = L La~. The.e 1 and.e oo norms are largest column and row sums of laij I.

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.

Outer product uv T
= column times row = rank one matrix.

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

Skewsymmetric matrix K.
The transpose is K, since Kij = Kji. Eigenvalues are pure imaginary, eigenvectors are orthogonal, eKt is an orthogonal matrix.

Spectrum of A = the set of eigenvalues {A I, ... , An}.
Spectral radius = max of IAi I.

Symmetric matrix A.
The transpose is AT = A, and aU = a ji. AI is also symmetric.

Tridiagonal matrix T: tij = 0 if Ii  j I > 1.
T 1 has rank 1 above and below diagonal.

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