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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 ∫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 ∫abf(x).dx=[∫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

∫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 F(b)−F(a) 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

∫abf(x).dx=[∫f(x).dx]ab.

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Most popular questions from this chapter

Calculate the exact value of each definite integral in Exercises 47–52 by using properties of definite integrals and the formulas in Theorem 4.13.

∫04(2x-3)2+5dx

Determine which of the limit of sums in Exercises 47–52 are infinite and which are finite. For each limit of sums that is finite, compute its value

limn→∞∑k=1nk2+k+1n2

Without calculating any sums or definite integrals, determine the values of the described quantities. (Hint: Sketch graphs first.)

(a) The signed area between the graph of f(x) = cos x and the x-axis on [−π, π].

(b) The average value of f(x) = cos x on [0, 2π].

(c) The area of the region between the graphs of f(x) =4−x2andg(x)=−4−x2on[−2,2].

Describe the intervals on which the function f is positive, negative, increasing and decreasing. Them describe the intervals on which the function A is positive , negative, increasing and decreasing

Write out all the integration formulas and rules that we know at this point.
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