Explain why or why not Determine whether the following statements are true and give an explanation or counterexample. Assume ƒ and ƒ' are continuous functions for all real numbers.

(d) If ƒ is continuous on [a,b] and ∫ₐᵇ |ƒ(𝓍)| d𝓍 = 0 , then ƒ(𝓍) = 0 on [a,b] .

Area by geometry Use geometry to evaluate the following definite integrals, where the graph of ƒ is given in the figure.

(a) ∫₀⁴ ƒ(𝓍) d𝓍

Function defined by an integral Let H (𝓍) = ∫₀ˣ √(4 ― t²) dt, for ― 2 ≤ 𝓍 ≤ 2.

(c) Evaluate H '(2) .

Evaluating integrals Evaluate the following integrals.

∫₀¹ 𝓍 • 2ˣ²⁺¹ d𝓍

Function defined by an integral Let H (𝓍) = ∫₀ˣ √(4 ― t²) dt, for ― 2 ≤ 𝓍 ≤ 2.

(a) Evaluate H (0) .

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample. Assume ƒ and ƒ' are continuous functions for all real numbers.

(g) ∫ ƒ' (g(𝓍))g' (𝓍) d(𝓍) = ƒ(g(𝓍)) + C .

(b) Find the average value of ƒ shown in the figure on the interval [2,6] and then find the point(s) c in (2, 6) guaranteed to exist by the Mean Value Theorem for Integrals. 

Evaluating integrals Evaluate the following integrals.

∫₋π/₂^π/² (cos 2𝓍 + cos 𝓍 sin 𝓍 ― 3 sin 𝓍⁵) 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.

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

Area by geometry Use geometry to evaluate the following definite integrals, where the graph of ƒ is given in the figure.

(c) ∫₅⁷ ƒ(𝓍) d𝓍

Evaluating integrals Evaluate the following integrals.                                                                                                                                         

 ∫ sin 𝒵 sin (cos 𝒵) d𝒵

Area by geometry Use geometry to evaluate the following definite integrals, where the graph of ƒ is given in the figure.

(b) ∫₆⁴ ƒ(𝓍) d𝓍

Integration by Riemann sums Consider the integral ∫₁⁴ (3𝓍― 2) d𝓍.

(a) Evaluate the right Riemann sum for the integral with n = 3 .

Evaluating integrals Evaluate the following integrals.

∫₋₅⁵ ω³ /√(ω⁵⁰ + ω²⁰ + 1) dω (Hint: Use symmetry . )

Area by geometry Use geometry to evaluate the following definite integrals, where the graph of ƒ is given in the figure.

(d) ∫₀⁷ ƒ(𝓍) d𝓍

Function defined by an integral Let H (𝓍) = ∫₀ˣ √(4 ― t²) dt, for ― 2 ≤ 𝓍 ≤ 2.

(e) Find the value of s such that H (𝓍) = sH(―𝓍)

Evaluating integrals Evaluate the following integrals.

∫√₂/₅^²/⁵ d𝓍/𝓍√(25𝓍² ―1)

Evaluating integrals Evaluate the following integrals.                                                                                                                                         

 ∫ (9𝓍⁸―7𝓍⁶) d𝓍

Use geometry and properties of integrals to evaluate the following definite integrals.​                                                                                          

 ∫₀⁴ √(8𝓍―𝓍²) d𝓍 . (Hint: Complete the square .)

Evaluating integrals Evaluate the following integrals.

∫₁⁴ ((√v + v) / v ) dv

Evaluating integrals Evaluate the following integrals.                                                                                                                                         

 ∫ 𝓍 sin 𝓍² cos⁸ 𝓍² d𝓍

Find the average value of ƒ(𝓍) = e²ˣ on [0, ln 2] .

Displacement from velocity A particle moves along a line with a velocity given by v(t) = 5 sin πt, starting with an initial position s(0) = 0 . Find the displacement of the particle between t = 0 and t = 2 , which is given by s(t) = ∫₀² v(t) dt . Find the distance traveled by the particle during this interval, which is ∫₀² |v(t)| dt .

Evaluating integrals Evaluate the following integrals.

∫π/₆^π/³ (sec² t + csc² t) dt

Integration by Riemann sums Consider the integral ∫₁⁴ (3𝓍― 2) d𝓍.

(a) Evaluate the right Riemann sum for the integral with n = 3 .

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample. Assume ƒ and ƒ' are continuous functions for all real numbers.

(a) A(𝓍) = ∫ₐˣ ƒ(t) dt and ƒ(t) = 2t―3 , then A is a quadratic 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.                                                                                                               

(a) Evaluate F(―2) and F(2).

Evaluating integrals Evaluate the following integrals.

∫₀¹ √𝓍 (√𝓍 + 1) d𝓍

Evaluating integrals Evaluate the following integrals.

∫₀^²π cos² 𝓍/6 d𝓍

Evaluate the following derivatives.

d/d𝓍 ∫₃ᵉˣ cos t² dt