8. Definite Integrals

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

∫₁⁴ (𝓍 ― 2)/√𝓍 d𝓍

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

∫π/₄^³π/⁴ (cot² 𝓍 + 1) d𝓍

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

∫₁² (z² + 4) / z dz

Derivatives of integrals Simplify the following expressions.

d/d𝓍 ∫₃ˣ (t² + t + 1) dt

Derivatives of integrals Simplify the following expressions.

d/dz ∫¹⁰ₛᵢₙ ₂ dt /(t⁴ + 1)

Derivatives of integrals Simplify the following expressions.

d/dy ∫¹⁰ᵧ³ √(𝓍⁶ + 1) d𝓍

Derivatives of integrals Simplify the following expressions.

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

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.

(b) Given an area function A(𝓍) = ∫ₐˣ ƒ(t) dt and an antiderivative F of ƒ, it follows that A'(𝓍) = F(𝓍) .

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.

(c) ∫ₐᵇ ƒ'(𝓍) d𝓍 = ƒ(b) ―ƒ(a) .

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)

Derivatives of integrals Simplify the following expressions.

d/d𝓍 ∫₀ˣ (√1 + t²) dt (Hint: ∫ˣ₋ₓ (√1 + t²) dt = ∫⁰₋ₓ (√1 + t²) dt + ∫ˣ₋ₓ (√1 + t²) dt ) .

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.

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.

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.

Suppose F is an antiderivative of ƒ and A is an area function of ƒ. What is the relationship between F and A?