Use geometry and properties of integrals to evaluate


∫₀¹ (2𝓍 + √(1―𝓍²) + 1) d𝓍

Definite integrals Use geometry (not Riemann sums) to evaluate the following definite integrals. Sketch a graph of the integrand, show the region in question, and interpret your result.

 ∫₀⁴ ƒ(𝓍) d𝓍, where ƒ(𝓍) = {5      if 𝓍 ≤ 2                                                                                                                                                                                     

                      3𝓍 ― 1  if 𝓍 > 2

Definite integrals Use a change of variables or Table 5.6 to evaluate the following definite integrals.                                                                                                                         

 ∫π/₄^π/² (cos 𝓍) / (sin² 𝓍) d𝓍

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.

(a) If ƒ is a constant function on the interval [a,b], then the right and left Riemann sums give the exact value of ∫ₐᵇ ƒ(𝓍) d𝓍, for any positive integer n.

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.

(b) If ƒ is a linear function on the interval [a,b] , then a midpoint Riemann sums give the exact value of ∫ₐᵇ ƒ(𝓍) d𝓍, for any positive integer n. 

Properties of integrals Suppose ∫₁⁴ ƒ(𝓍) d𝓍 = 6 , ∫₁⁴ g(𝓍) d𝓍 = 4 and ∫₃⁴ ƒ(𝓍) d𝓍 = 2 . Evaluate the following integrals or state that there is not enough information.

―∫₄¹ 2ƒ(𝓍) d𝓍

Properties of integrals Suppose ∫₁⁴ ƒ(𝓍) d𝓍 = 6 , ∫₁⁴ g(𝓍) d𝓍 = 4 and ∫₃⁴ ƒ(𝓍) d𝓍 = 2 . Evaluate the following integrals or state that there is not enough information.

∫₁³ ƒ(𝓍)/g(𝓍) d𝓍

Evaluating integrals Evaluate the following integrals.

∫₋₂² (3𝓍⁴―2𝓍 + 1) d𝓍

Evaluating integrals Evaluate the following integrals.

∫₀¹ √𝓍 (√𝓍 + 1) d𝓍