Sigma notation Express the following sums using sigma notation. (Answers are not unique.)
(a) 1 + 2 + 3 + 4 + 5

Area functions for the same linear function Let ƒ(t) = t and consider the two area functions A(𝓍) = ∫₀ˣ ƒ(t) dt and F(𝓍) = ∫₂ˣ ƒ(t) dt .

(b) Evaluate F(4) and F(6). Then use geometry to find an expression for F (𝓍) , for 𝓍 ≥ 2. 

Suppose ƒ is an even function and ∫⁸₋₈ ƒ(𝓍) d𝓍 = 18

(b) Evaluate ∫₋₈⁸ 𝓍ƒ(𝓍) d𝓍 .

Bounds on an integral Suppose ƒ is continuous on [a, b] with ƒ''(𝓍) > 0 on the interval. It can be shown that (b―a) ƒ [(a + b) /2] ≤ ∫ₐᵇ ƒ(𝓍) d𝓍 ≤ (b―a) [ (ƒ(a) + ƒ(b)) /2]                                                         

(a) Assuming ƒ is nonnegative on [a, b], draw a figure to illustrate the geometric meaning of these inequalities. Discuss your conclusions. b. 

Working with area functions Consider the function ƒ and its graph.

(a) Estimate the zeros of the area function A(𝓍) = ∫₀ˣ ƒ(t) dt , for 0 ≤ 𝓍 ≤ 10 .

Using properties of integrals Use the value of the first integral I to evaluate the two given integrals. 

I = ∫₀^π/2 (cos θ ― 2 sin θ) dθ = ―1

(a) ∫₀^π/2 (2 sin θ ― cos θ) dθ

The following functions are positive and negative on the given interval.

ƒ(𝓍) = xe⁻ˣ on [-1,1]

(b) Approximate the net area bounded by the graph of f and the x-axis on the interval using a left, right, and midpoint Riemann sum with n = 4.