BackExplain 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.
Use geometry and properties of integrals to evaluate
∫₀¹ (2𝓍 + √(1―𝓍²) + 1) 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.
―∫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
Evaluating integrals Evaluate the following integrals.
∫π/₆^π/³ (sec² t + csc² t) dt
Evaluating integrals Evaluate the following integrals.
∫₁⁴ ((√v + v) / v ) dv
Evaluating integrals Evaluate the following integrals.
∫₋₅⁵ ω³ /√(ω⁵⁰ + ω²⁰ + 1) dω (Hint: Use symmetry . )
Evaluating integrals Evaluate the following integrals.
∫₋π/₂^π/² (cos 2𝓍 + cos 𝓍 sin 𝓍 ― 3 sin 𝓍⁵) d𝓍
Evaluating integrals Evaluate the following integrals.
∫₀⁵ |2𝓍―8|d𝓍
Definite integrals Use a change of variables or Table 5.6 to evaluate the following definite integrals.
∫₀¹ 2e²ˣ d𝓍
Function defined by an integral Let H (𝓍) = ∫₀ˣ √(4 ― t²) dt, for ― 2 ≤ 𝓍 ≤ 2.
(a) Evaluate H (0) .
Function defined by an integral Let H (𝓍) = ∫₀ˣ √(4 ― t²) dt, for ― 2 ≤ 𝓍 ≤ 2.
(e) Find the value of s such that H (𝓍) = sH(―𝓍)