8. Definite Integrals

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.

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.

(c) Divide [0, 2] into n = 4 subintervals and approximate the area of the region using a right Riemann sum. Illustrate the solution geometrically.

Estimate ∫₁⁴ √(4𝓍 + 1) d𝓍 by evaluating the left, right, and midpoint Riemann sums using a regular partition with n = 6 subintervals.