Evaluating integrals Evaluate the following integrals.


∫π/₁₂^π/⁹ (csc 3𝓍 cot 3𝓍 + sec 3𝓍 tan 3𝓍) 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.

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

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 .

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.

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

Evaluating integrals Evaluate the following integrals.                                                                                                                                         

 ∫ y² (3y³ + 1)⁴ dy

Estimate ∫₁⁴ √(4𝓍 + 1) d𝓍 by evaluating the left, right, and midpoint Riemann sums using a regular partition with n = 6 subintervals.