8. Definite Integrals

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

(b) Graph ƒ and A.

ƒ(𝓍) = 1/𝓍 ; a = 1 , b = 4 , c = 6

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.                                                                                                               

(a) Evaluate F(―2) and F(2).

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.

(f) Find a constant C such that F(𝓍) = G(𝓍) + C .

Derivatives of integrals Simplify the following expressions.

d/dz ∫¹⁰ₛᵢₙ ₂ dt /(t⁴ + 1)

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.

Find dy/dx if y = ∫(From cos x to 0) 1/(1 - t²) dt.

Explain the main steps in your calculation.

Evaluate the integrals in Exercises 47–68.

∫⁰-π/3 sec x tan x dx

Evaluate the integrals in Exercises 111–114.

113. ∫₁^(1/x) (1 / t) dt,x > 0

Evaluate ∫₃⁸ ƒ ′(t) dt , where ƒ ′ is continuous on [3, 8], ƒ(3) = 4, and ƒ(8) = 20 .

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

∫₁⁹ 2/(√𝓍) d𝓍

Given the definite integral $F\left(x\right)={\int }_3^x[t^8-\sin \left(t^4\right)]dt$ , find the derivative $F^{\prime }\left(x\right)$.

Area functions The graph of ƒ is shown in the figure. Let A(x) = ∫₀ˣ ƒ(t) dt and F(x) = ∫₂ˣ ƒ(t) dt be two area functions for ƒ. Evaluate the following area functions.

(g) F(2)

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. 

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.

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

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.