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 .)

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

(c) Evaluate H '(2) .

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.

Evaluating integrals Evaluate the following integrals.                                                                                                                                         

 ∫ y² (3y³ + 1)⁴ dy