8. Definite Integrals

Find the average value $f_{ave}$ of the function $f\left(x\right)=3x^2+8x$ on the interval $[-1,5]$.

Symmetry of composite functions Prove that the integrand is either even or odd. Then give the value of the integral or show how it can be simplified. Assume f and g are even functions and p and q are odd functions.

∫ᵃ₋ₐ ƒ(g(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. 

Let $f\left(x\right)=x^{\frac{1}{2}}$. What is the average value of $f$ on the interval $[0,16]$?

Average Value: Find the average value of the function f(x) = 1 / (1 - sin θ) on the interval [0, π/6].

Average distance on a parabola What is the average distance between the parabola y = 30𝓍 (20 ― 𝓍 ) and the 𝓍-axis on the interval [0, 20] ?

What is the average value of the function $f\left(x\right)=x^2$ on the interval $[0,2]$?

Suppose ƒ is an odd function, ∫₀⁴ ƒ(𝓍) d𝓍 = 3 , and ∫₀⁸ ƒ(𝓍) d𝓍 = 9 .

(a) Evaluate ∫₋₈⁴ ƒ(𝓍) d𝓍 .

Find the average value $f_{ave}$ of the function $f\left(t\right)=e^{\tan \left(t\right)}\tan \left(t\right)$ on the interval $[0,\frac{\pi }{2}]$.

If ƒ is an even function, why is ∫ᵃ₋ₐ ƒ(𝓍) d𝓍 = 2 ∫₀ᵃ ƒ(𝓍) d𝓍?

Average value with a parameter Consider the function ƒ(𝓍) = a𝓍 (1―𝓍) on the interval [0, 1], where a is a positive real number.

(a) Find the average value of ƒ as a function of a .

Find the average value of f(x) = (√(x + 1)) / √x on the interval [1, 3].

Gateway Arch The Gateway Arch in St. Louis is 630 ft high and has a 630-ft base. Its shape can be modeled by the parabola y = 630 (1― (𝓍/315)²) . Find the average height of the arch above the ground.

What is the average value of the function $f\left(x\right)=x^2$ on the interval $[0,2]$?

Given a function $f\left(x\right)$, what is the average rate of change of $f\left(x\right)$ over the interval $[4,13]$?