Estimate ¹ cos(x²) dx using (a) the Trapezoidal Rule and (b) the Midpoint Rule, each with n = 4. Give each answer correct to five decimal places. (a) T4 = (b) M4 = (c) By looking at a sketch of the graph of the integrand, determine for each estimate whether it overestimates, underestimates, or is the exact area. 1. M4 Overestimate Underestimate 2. T4 (d) What can you conclude about the true value of the integral? A. No conclusions can be drawn. B. T₁ < ₁²¹ cos(x²) dx and M₁ < C. T4 > focos (x²) dx and M₁ > D. M₁
Estimate ¹ cos(x²) dx using (a) the Trapezoidal Rule and (b) the Midpoint Rule, each with n = 4. Give each answer correct to five decimal places. (a) T4 = (b) M4 = (c) By looking at a sketch of the graph of the integrand, determine for each estimate whether it overestimates, underestimates, or is the exact area. 1. M4 Overestimate Underestimate 2. T4 (d) What can you conclude about the true value of the integral? A. No conclusions can be drawn. B. T₁ < ₁²¹ cos(x²) dx and M₁ < C. T4 > focos (x²) dx and M₁ > D. M₁
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Transcribed Image Text:**Estimating \(\int_0^1 \cos(x^2) \, dx\) Using Numerical Methods**
Estimate \(\int_0^1 \cos(x^2) \, dx\) using (a) the Trapezoidal Rule and (b) the Midpoint Rule, each with \(n = 4\). Give each answer correct to five decimal places.
**(a) Trapezoidal Rule (T₄):**
- Enter the value in the box provided: \(\boxed{x^2}\)
**(b) Midpoint Rule (M₄):**
- Enter the value in the box provided: \(\boxed{\phantom{x}}\)
**(c) Analyzing Estimates:**
By looking at a sketch of the graph of the integrand, determine for each estimate whether it overestimates, underestimates, or is the exact area.
- **1. \(M_4\):** Overestimate
- **2. \(T_4\):** Underestimate
**(d) Conclusion About the True Value of the Integral:**
What can you conclude about the true value of the integral?
- **A.** No conclusions can be drawn.
- **B.** \(T_4 < \int_0^1 \cos(x^2) \, dx\) and \(M_4 < \int_0^1 \cos(x^2) \, dx\)
- **C.** \(T_4 > \int_0^1 \cos(x^2) \, dx\) and \(M_4 > \int_0^1 \cos(x^2) \, dx\)
- **D.** \(M_4 < \int_0^1 \cos(x^2) \, dx < T_4\)
- **E.** \(\boxed{T_4 < \int_0^1 \cos(x^2) \, dx < M_4}\) (Correct Answer)
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