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

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

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) = 3t + 1 , a = 2

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 .                                                                                                                               

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 ƒ(t) = 4t + 2 , a = 0

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.

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.

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).

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

Definite integrals Evaluate the following integrals using the Fundamental Theorem of Calculus. Explain why your result is consistent with the figure.

∫₀¹ (𝓍² ― 2𝓍 + 3) d𝓍

{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) 

{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

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.

Evaluate

lim [ ∫₂ˣ √(t² + t + 3dt) ] / (𝓍² ―4)

𝓍→2

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.