Use geometry and properties of integrals to evaluate the following definite integrals.
∫₄⁰ (2𝓍 + √(16―𝓍²)) d𝓍 . (Hint: Write the integral as sum of two integrals.)

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

(a) Evaluate H (0) .

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.

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

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

(c) Evaluate the definite integral by taking the limit as n →∞ of the Riemann sum in part (b).

Evaluating integrals Evaluate the following integrals.                                                                                                                                         

 ∫ y² /(y³ + 27) dy

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.

(f) ∫ₐᵇ (2 ƒ(𝓍) ―3g (𝓍)) d𝓍 = 2 ∫ₐᵇ ƒ(𝓍) d𝓍 + 3 ∫₆ᵃ g(𝓍) d𝓍 .

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) .