Use Romberg integration to compute R3,3 for the following integrals. a. 1.5 1 x2 ln x dx

Use Romberg integration to compute \(R_{3,3}\) for the following integrals. a. \(\int_{1}^{1.5} x^{2} \ln x d x\) b. \(\int_{0}^{1} x^{2} e^{-x} d x\) c. \(\int_{0}^{0.35} \frac{2}{x^{2}-4} d x\) d. \(\int_{0}^{\frac{\pi}{4}} x^{2} \sin x d x\) e. \(\int_{0}^{\frac{\pi}{4}} e^{3 x} \sin 2 x d x\) f. \(\int_{1}^{1.6} \frac{2 x}{\left(x^{2}-4\right)} d x\) g. \(\int_{3}^{3.5} \frac{x}{\sqrt{\left(x^{2}-4\right)}} d x\) h. \(\int_{0}^{\frac{\pi}{4}}(\cos x)^{2} d x\).

Use Romberg integration to compute R3,3 for the following integrals. a. 1 1 (cos x) 2 dx b. 0.75 0.75 x ln(x + 1) dx c. 4 1  (sin x) 2 2x sin x + 1  dx d. 2e e 1 x ln x dx 3.

Calculate R4,4 for the integrals in Exercise 1.

Calculate R4,4 for the integrals in Exercise 2.

Use Romberg integration to approximate the integrals in Exercise 1 to within 106. Compute the Romberg table until either |Rn1,n1 Rn,n| < 106, or n = 10. Compare your results to the exact values of the integrals.

Use Romberg integration to approximate the integrals in Exercise 2 to within 106. Compute the Romberg table until either |Rn1,n1 Rn,n| < 106, or n = 10. Compare your results to the exact values of the integrals.

Use the following data to approximate  5 1 f (x) dx as accurately as possible. x 1 2 3 4 5 f (x) 2.4142 2.6734 2.8974 3.0976 3.2804

Romberg integration is used to approximate 1 0 x2 1 + x3 dx. If R11 = 0.250 and R22 = 0.2315, what is R21?

Romberg integration is used to approximate 3 2 f (x) dx. If f (2) = 0.51342, f (3) = 0.36788, R31 = 0.43687, and R33 = 0.43662, find f (2.5).

Romberg integration for approximating  1 0 f (x) dx gives R11 = 4 and R22 = 5. Find f (1/2).

Romberg integration for approximating  b a f (x) dx gives R11 = 8, R22 = 16/3, and R33 = 208/45. Find R31.

Use Romberg integration to compute the following approximations to 48 0  1 + (cos x)2 dx. [Note: The results in this exercise are most interesting if you are using a device with between sevenand nine-digit arithmetic.] a. Determine R1,1, R2,1, R3,1, R4,1, and R5,1, and use these approximations to predict the value of the integral. b. Determine R2,2, R3,3, R4,4, and R5,5, and modify your prediction. c. Determine R6,1, R6,2, R6,3, R6,4, R6,5, and R6,6, and modify your prediction. d. Determine R7,7, R8,8, R9,9, and R10,10, and make a final prediction. e. Explain why this integral causes difficulty with Romberg integration and how it can be reformulated to more easily determine an accurate approximation.

Show that the approximation obtained from Rk,2 is the same as that given by the Composite Simpsons rule described in Theorem 4.4 with h = hk .

Show that, for any k, 2k 11 i=1 f a + i 2 hk1  = 2 k2 i=1 f a + i 1 2  hk1  + 2k 21 i=1 f (a + ihk1).

Use the result of Exercise 14 to verify Eq. (4.34); that is, show that for all k, Rk,1 = 1 2 Rk1,1 + hk1 2 k2 i=1 f a + i 1 2  hk1  .

In Exercise 26 of Section 1.1, a Maclaurin series was integrated to approximate erf(1), where erf(x) is the normal distribution error function defined by erf(x) = 2 x 0 et 2 dt. Approximate erf(1) to within 107.