91Ó°ÊÓ

Notation: Describe the meanings of each of the following mathematical expressions or how they are commonly used in this chapter:

$f\left({x_k}^{∗}\right)∆x$

Use evaluation notation and the Fundamental Theorem of Calculus to prove Theorem 4.26:

${∫}_a^{b}f\left(x\right)dx={\left[∫f\left(x\right)dx\right]}_a^b$

In this exercise you will use two different methods to prove that, for any real numbers a, b, c, and k, ${∫}_{-k}^k\left(a{x}^2+c\right)dx=2{∫}_0^k\left(a{x}^2+c\right)dx.$

(a) Prove this equality by using a geometric argument that involves signed area.

(b) Now prove the equality a different way, by using an algebraic argument and the Fundamental Theorem of Calculus.

Suppose f is an integrable function on [a, b] and $x_k=a+k\left(\frac{b-a}{n}\right).$

(a) Use the definition of the definite integral as a limit of Riemann sums to show that

$\underset{n→∞}{\lim }\frac{\underset{k=1}{\overset{n}{∑}}f\left(x_k\right)}{n}=\frac{1}{b-a}{∫}_a^{b}f\left(x\right)dx.$

(b) Why is it algebraically sensible that the left-hand side of the equation is a calculation of average value?

(c) Why is it graphically sensible that the right-hand side of the equation is a calculation of average value?

Prove the Mean Value Theorem for Integrals by following these steps:

(a) Use the Extreme Value Theorem to argue that f has a maximum value M and a minimum value m on the interval [a, b].

(b) Use an upper sum and a lower sum with one rectangle to argue that $m(b-a)â\copyright ½{∫}_a^{b}f\left(x\right)dxâ\copyright ½M(b-a)$and thus that the average value of f on [a, b] is between m and M.

Explain why the upper sum approximation for the area

between the graph of a function f and the x-axis on [a, b]

must always be larger than or equal to any other type of

Riemann sum approximation with the same number n of

rectangles.

Explain why at this point we don’t have an integration formula for the function$f\left(x\right)=secx$ whereas we do have an integration formula for $f\left(x\right)=\sin x$.

Consider the sum$\underset{k=m}{\overset{n}{∑}}{a}_k=9+16+25+36+49$

Riemann sums: Calculate each of the following Riemann sum

approximations for the definite integral of f on [a, b], using the

given value of n.

The midpoint sum for$f\left(x\right)=9-x^2$on$\left[0,3\right],n=3$.

The function,

$\ln x={∫}_1^x\frac{1}{t}dt$

is continuous and differentiable on the interval ____ .