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Chapter 4: Q. 4 (page 372)

State the Fundamental Theorem of Calculus

(a) in its original form,

(b) in its alternative form, and

(c) by using an indefinite integral and evaluation notation.

Short Answer

Expert verified

(a) The fundamental Theorem of calculus in its original form is 鈭�abf(x).dx=F(b)-F(a).

(b) The fundamental Theorem of calculus in its alternative form is[F(x)]ab=F(b)-F(a).

(c) The fundamental Theorem of calculus in its by using an indefinite integral and evaluation notation is 鈭�abf(x).dx=[鈭�f(x).dx]ab.

Step by step solution

01

Part (a) Step 1. Given Information.

The fundamental theorem of calculus.

02

Part (a) Step 2. Original form.

If f is continuous on [a, b] and F is any antiderivative of f, then

鈭�abf(x).dx=F(b)-F(a)

03

Part (b) Step 1. Alternative form.

For any function F on an interval [a, b], the difference 贵(产)鈭扚(补) will be called the evaluation of F(x) on[a, b] and will be denoted by [F(x)]ab=F(b)-F(a).

04

Part (c) Step 1. Evaluation Notation.

If f is continuous on [a, b], then

鈭�abf(x).dx=[鈭�f(x).dx]ab.

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Most popular questions from this chapter

Find the sum or quantity without completely expanding or calculating any sums.

Given鈭�k=025k=325and鈭�k=328k-32=14,910, find the value of鈭�k=325k2-5k+9.

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鈭�(sec2x+csc2x)dx.

For each function f and interval [a, b] in Exercises 27鈥�33, use the given approximation method to approximate the signed area between the graph of f and the x-axis on [a, b]. Determine whether each of your approximations is likely to be an over-approximation or an under-approximation of the actual area.

f(x)=x2,[a,b]=[0,3]left sum with

a) n = 3 b) n = 6

Use integration formulas to solve each integral in Exercises 21鈥�62. You may have to use algebra, educated guess and-check, and/or recognize an integrand as the result of a product, quotient, or chain rule calculation. Check each of your answers by differentiating.

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鈭�-2-2x(f(x)+3)2dx
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