BackExplain why or why not Determine whether the following statements are true and give an explanation or counterexample.
(a) Consider the linear function ƒ(𝓍) = 2x + 5 and the region bounded by its graph and the x-axis on the interval [3,6]. Suppose the area of this region is approximated using midpoint Riemann sums. Then the approximations give the exact area of the region for any number of subintervals.
Integration by Riemann sums Consider the integral ∫₁⁴ (3𝓍― 2) d𝓍.
(a) Evaluate the right Riemann sum for the integral with n = 3 .
Integration by Riemann sums Consider the integral ∫₁⁴ (3𝓍― 2) d𝓍.
(b) Use summation notation to express the right Riemann sum in terms of a positive integer n .
Integration by Riemann sums Consider the integral ∫₁⁴ (3𝓍― 2) d𝓍.
(c) Evaluate the definite integral by taking the limit as n →∞ of the Riemann sum in part (b).
Integration by Riemann sums Consider the integral ∫₁⁴ (3𝓍― 2) d𝓍.
(a) Evaluate the right Riemann sum for the integral with n = 3 .
Limit definition of the definite integral Use the limit definition of the definite integral with right Riemann sums and a regular partition to evaluate the following definite integrals. Use the Fundamental Theorem of Calculus to check your answer.
∫₀² (𝓍²―4) d𝓍
Limit definition of the definite integral Use the limit definition of the definite integral with right Riemann sums and a regular partition to evaluate the following definite integrals. Use the Fundamental Theorem of Calculus to check your answer.
∫₀⁴ (𝓍³―𝓍) d𝓍
3. Explain geometrically how the Trapezoid Rule is used to approximate a definite integral.
5-8. Compute the following estimates of ∫(0 to 8) f(x) dx using the graph in the figure.
6. T(4)
9. If the Trapezoid Rule is used on the interval [-1, 9] with n = 5 subintervals, at what x-coordinates is the integrand evaluated?
15-18. {Use of Tech} Midpoint Rule approximations. Find the indicated Midpoint Rule approximations to the following integrals.
15. ∫(2 to 10) 2x² dx using n = 1, 2, and 4 subintervals
15-18. {Use of Tech} Midpoint Rule approximations. Find the indicated Midpoint Rule approximations to the following integrals.
18. ∫(0 to 1) e⁻ˣ dx using n = 8 subintervals
19-22. {Use of Tech} Trapezoid Rule approximations. Find the indicated Trapezoid Rule approximations to the following integrals.
21. ∫(0 to 1) sin(πx) dx using n = 6 subintervals
23-26. {Use of Tech} Simpson's Rule approximations. Find the indicated Simpson's Rule approximations to the following integrals.
24. ∫(4 to 8) √x dx using n = 4 and n = 8 subintervals
Approximating areas Estimate the area of the region bounded by the graph of ƒ(𝓍) = x² + 2 and the x-axis on [0, 2] in the following ways.
(a) Divide [0, 2] into n = 4 subintervals and approximate the area of the region using a left Riemann sum. Illustrate the solution geometrically.