8. Definite Integrals

Area functions for constant functions Consider the following functions ƒ and real numbers a (see figure).

(b) Verify that .A'(𝓍) = ƒ(𝓍)

ƒ(t) = 5 , a = -5

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)

Area functions from graphs The graph of ƒ is given in the figure. A(𝓍) = ∫₀ˣ ƒ(t) dt and evaluate A(2), A(5), A(8), and A(12).

Definite integrals Evaluate the following integrals using the Fundamental Theorem of Calculus

∫₁⁴ (𝓍 ― 2)/√𝓍 d𝓍

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.

Area functions for constant functions Consider the following functions ƒ and real numbers a (see figure).

(a) Find and graph the area function A(𝓍) = ∫ₐˣ ƒ(t) dt for ƒ.

ƒ(t) = 5 , a = 0

For Exercises 127 and 128 find a function f satisfying each equation.

127. ∫₂ˣ √(f(t)) dt = x ln x

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

 ∫₋π^π cos² 𝓍 d𝓍

Working with area functions Consider the function ƒ and the points a, b, and c.

(c) Evaluate A(b) and A(c). Interpret the results using the graphs of part (b) .

ƒ(𝓍) = ― 12𝓍 (𝓍―1) (𝓍― 2) ; a = 0 , b = 1 , c = 2

Limits with integrals Evaluate the following limits.

lim ∫₂ˣ eᵗ² dt

𝓍→2 ---------------

𝓍 ― 2

Determine the intervals on which the function g(𝓍) = ∫ₓ⁰ t / (t² + 1) dt  is concave up or concave down.

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 .

Evaluate

lim [ ∫₂ˣ √(t² + t + 3dt) ] / (𝓍² ―4)

𝓍→2

Definite integrals Evaluate the following integrals using the Fundamental Theorem of Calculus

∫₁² (z² + 4) / z dz

Working with area functions Consider the function ƒ and the points a, b, and c.

(a) Find the area function A (𝓍) = ∫ₐˣ ƒ(t) dt using the Fundamental Theorem.

ƒ(𝓍) = sin 𝓍 ; a = 0 , b = π/2 , c = π