8. Definite Integrals

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) = cos 𝓍 ; a = 0 , b = π/2 , c = π

Area functions and the Fundamental Theorem Consider the function

ƒ(t) = { t      if  ―2 ≤ t < 0

t²/2    if    0 ≤ t ≤ 2

and its graph shown below. Let F(𝓍) = ∫₋₁ˣ ƒ(t) dt and G(𝓍) = ∫₋₂ˣ ƒ(t) dt.

(b) Use the Fundamental Theorem to find an expression for F '(𝓍) for ―2 ≤ 𝓍 < 0.

Evaluate the integrals in Exercises 111–114.

112. ∫₁^(eˣ) (1 / t) dt

Working with area functions Consider the function ƒ and its graph.

(b) Estimate the points (if any) at which A has a local maximum or minimum.

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

∫π/₄^³π/⁴ (cot² 𝓍 + 1) d𝓍

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?

Working with area functions Consider the function ƒ and its graph.

(a) Estimate the zeros of the area function A(𝓍) = ∫₀ˣ ƒ(t) dt , for 0 ≤ 𝓍 ≤ 10 .

Evaluate the following integral:

${\int }_{-1}^2(x^2-3x+2)dx$

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 

Area functions and the Fundamental Theorem Consider the function

ƒ(t) = { t      if  ―2 ≤ t < 0

t²/2    if    0 ≤ t ≤ 2

and its graph shown below. Let F(𝓍) = ∫₋₁ˣ ƒ(t) dt and G(𝓍) = ∫₋₂ˣ ƒ(t) dt.

(e) Evaluate F ''(―1) and F ''(1). Interpret these values.

Evaluate the following integral:

${\int }_1^4(2x+1)dx$

Matching functions with area functions Match the functions ƒ, whose graphs are given in a― d, with the area functions A (𝓍) = ∫₀ˣ ƒ(t) dt, whose graphs are given in A–D.

Explain why ∫ₐᵇ ƒ ′(𝓍) d𝓍 = ƒ(b) ― ƒ(a)

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.

Given the definite integral $F\(\left\)(x\(\right\))=\(\int\)_{12}^{20x}\!\(\left\)(h^4+\(\frac{63h}{\sqrt{h^5}\)}\(\right\))\,dh$, find the derivative $F^{\(\prime\)}\(\left\)(x\(\right\))$.