8. Definite Integrals

Max/min of area functions Suppose ƒ is continuous on [0 ,∞) and A(𝓍) is the net area of the region bounded by the graph of ƒ and the t-axis on [0, x]. Show that the local maxima and minima of A occur at the zeros of ƒ. Verify this fact with the function ƒ(𝓍) = 𝓍² - 10𝓍.

Area functions for linear 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 .

ƒ(t) = 2t + 5 , a = 0

In Exercises 75–78, find dy/dx.

y = ∫(from x to 1) (6/(3 + t^4))dt

Evaluate the integrals in Exercises 111–114.

111. ∫₁^(ln x) (1 / t) dt,x > 1

Area functions for linear 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 .                                                                                                                               

 <IMAGE>                                                                                                                                                                                                           

 ƒ(t) = 4t + 2 , a = 0

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

(b) Graph ƒ and A.

ƒ(𝓍) = eˣ ; a = 0 , b = ln 2 , c = ln 4

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

b) Verify that A'(𝓍) = ƒ(𝓍).

ƒ(t) = 4t + 2 , a = 0

Derivatives of integrals Simplify the following expressions.

d/dt ∫₀ᵗ d𝓍/(1 + 𝓍²) + ∫₁¹/ᵗ dx/(1 + 𝓍²)

Suppose the graph of a continuous function $f$ is shown below, and the area between the graph of $f$ and the $x$-axis from $x=0$ to $x=2$ is $3$ (above the $x$-axis), and from $x=2$ to $x=4$ is $2$ (below the $x$-axis). What is the value of the definite integral ${\int }_0^{4}f\left(x\right)dx$?

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

∫₁⁸ 8𝓍¹/³ d𝓍

Evaluate the integrals in Exercises 47–68.

∫₀^π/3 sec² θ dθ

{Use of Tech} Functions defined by integrals Consider the function g, which is given in terms of a definite integral with a variable upper limit.

b) Calculate g'(𝓍)

g(𝓍) = ∫₀ˣ sin² t 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.

{Use of Tech} Functions defined by integrals Consider the function g, which is given in terms of a definite integral with a variable upper limit.

(b) Calculate g'(𝓍)

g(𝓍) = ∫₀ˣ sin (πt² ) dt ( a Fresnel integral) 

Evaluate the following integral:

${\int }_0^{3}4dx$