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Chapter 4: Q. 55 (page 353)

Use the definition of the definite integral as a limit of Riemann sums to prove Theorem 4.11(b): For any function f that is integrable on [a,b] and any real number c,

鈭�abcf(x)dx=c鈭�abf(x)dx

Short Answer

Expert verified

The theorem 4.11(b) is proved.

鈭�abcf(x)dx=c鈭�abf(x)dx

Step by step solution

01

Step 1. Given Information

We are given a function f that is integrable.

02

Step 2. Proving the theorem

Proving the theorem,

鈭�abcf(x)=limn鈫�鈭�鈭�k=1ncfxk*螖x=climn鈫�鈭�鈭�k=1nfxk*螖x=c鈭�abf(x)

Hence Proved.

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Most popular questions from this chapter

If 鈭�-23f(x)dx=4,鈭�-26f(x)dx=9,鈭�-23g(x)dx=2and 鈭�36g(x)dx=3,then find the values of each definite integral in Exercises 29-40. If there is not enough information, explain why.

鈭�36fxdx.

Write each expression in Exercises 41鈥�43 in one sigma notation (with some extra terms added to or subtracted from the sum, as necessary).

3鈭�k=225k2+2鈭�kk=224-鈭�k=0251

If f is negative on [鈭�3, 2], is the definite integral 鈭�-32f(x)dx positive or negative? What about the definite integral 鈭� 鈭�-32f(x)dx?

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

limn鈫�鈭�鈭�k=1nk2+k+1n2

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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