8. Definite Integrals

Function defined by an integral Let H (𝓍) = ∫₀ˣ √(4 ― t²) dt, for ― 2 ≤ 𝓍 ≤ 2.

(c) Evaluate H '(2) .

Limits with integrals Evaluate the following limits.

lim ∫₂ˣ eᵗ² dt

𝓍→2 ---------------

𝓍 ― 2

Function defined by an integral Let ƒ(𝓍) = ∫₀ˣ (t ― 1)¹⁵ (t―2)⁹ dt .

(c) For what values of 𝓍 does ƒ have local minima? Local maxima?

Integrals with sin² 𝓍 and cos² 𝓍 Evaluate the following integrals.                                                                                                             

 ∫₀^π/⁴ cos² 8θ dθ

Integrals with sin² 𝓍 and cos² 𝓍 Evaluate the following integrals.                                                                                                             

 ∫₋π^π cos² 𝓍 d𝓍

Definite integrals Evaluate the following integrals using the Fundamental Theorem of Calculus. Sketch the graph of the integrand and shade the region whose net area you have found.                       

 ∫₀⁵ (𝓍²―9) d𝓍 

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

                                                                                                                                                                                     (b) Suppose ƒ is a negative increasing function, for 𝓍 > 0 . Then the area function A(𝓍) = ∫₀ˣ ƒ(t) dt is a decreasing function of 𝓍 .