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Chapter 4: Q. 57 (page 353)

Use the definition of the definite integral as a limit of Riemann sums to prove Theorem 4.12(b): For any function f that is integrable on [a,b]

鈭�baf(x)dx=-鈭�abf(x)dx

Short Answer

Expert verified

The theorem 4.12(b) is proved.

鈭�baf(x)dx=-鈭�abf(x)dx

Step by step solution

01

Step 1. Given Information 

We are given a functionf that is integrable.

02

Step 2. Proving the theorem

On the interval [a,b], the value of 螖xis,

螖x=b-an

And on the interval $[b, a]$, the value of 螖xis,

螖x'=a-bn=-b-an=-螖x

03

Step 3. Proving the theorem

Proving the theorem,

鈭�baf(x)dx=limn鈫�鈭�鈭�k=1nfxk*螖x*=limn鈫�鈭�鈭�k=1nfxk*-螖x=-鈭�abf(x)dx

Hence Proved.

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Most popular questions from this chapter

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. (Hint for Exercise 54: tanx=sinxcosx).

鈭�2xlnx-xlnx2dx.

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. (Hint for Exercise 54: tanx=sinxcosx).

鈭�(tanx+xsec2x)dx.

Use the Fundamental Theorem of Calculus to find the exact values of the given definite integrals. Use a graph to check your answer.

鈭�01鈥�x1+x2dx

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.

Show by exhibiting a counterexample that, in general, 鈭�f(x)g(x)dx鈮�鈭�f(x)dx鈭�g(x)dx. In other words, find two functions f and g so that the integral of their product is not equal to the product of their integrals.
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