8. Definite Integrals

Evaluate the following derivatives.

d/d𝓍 ∫₃ᵉˣ cos t² dt

Find the intervals on which ƒ(𝓍) = ∫ₓ¹ (t―3) (t―6)¹¹ dt is increasing and the intervals on which it is decreasing.

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 .

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

                                                                                                                                                                                     (c) The functions p(𝓍) = sin 3𝓍 and q(𝓍) = 4 sin 3𝓍 are antiderivatives of the same function. 

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

                                                                                                                                                                                     (d) If A(𝓍) = 3𝓍²― 𝓍― 3 is an area function for ƒ, then                                                                                                                                   

     B(𝓍) = 3𝓍² ― 𝓍 is also an area function for ƒ.

Suppose f and g have continuous derivatives on an interval [a, b]. Prove that if f(a)=g(a) and f(b)=g(b), then ∫a^b f′(x) dx = ∫a^b g′(x) dx.

Cubic zero net area Consider the graph of the cubic y = 𝓍 (𝓍― a) (𝓍― b), where 0 < a < b. Verify that the graph bounds a region above the 𝓍-axis, for 0 < 𝓍 < a , and bounds a region below the 𝓍-axis, for a < 𝓍 < b. What is the relationship between a and b if the areas of these two regions are equal? 

{Use of Tech} Fresnel integrals The theory of optics gives rise to the two Fresnel integrals

S(x) = ∫₀ˣ sin t² dt and C(x) = ∫₀ˣ cos t² dt

a. Compute S′(x) and C′(x).

Generalizing the Mean Value Theorem for Integrals Suppose ƒ and g are continuous on [a, b] and let h(𝓍) = (𝓍―b) ∫ₐˣ ƒ(t) dt + (𝓍―a) ∫ₓᵇg(t)dt.                                                                                                                                                                                                                                                                                                                                

(b) Show that there is a number c in (a, b) such that ∫ₐᶜ ƒ(t) dt = ƒ(c) (b ― c)                                                                                                              

(Source: The College Mathematics Journal, 33, 5, Nov 2002)