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Chapter 4: Q. 58 (page 353)

Give a geometric argument to prove Theorem 4.13(a): For any real numbers a, b, and c,

鈭�abcdx=c(b-a)

(Hint: Use a rectangle.)

Short Answer

Expert verified

The theorem 4.13(a) is proved.

鈭�abcdx=c(b-a)

Step by step solution

01

Step 1. Given Information 

We are given a theorem,

鈭�abcdx=c(b-a)

02

Step 2. Proving the theorem 

Take, f(x)=con [a,b]. Then the integral 鈭�abf(x)dx, represents the area from a to b under the constant curve f(x)=c. That is, the integral 鈭�abf(x)dx, represents the area of the rectangle of length (b-a) and breadth c.

Hence Proved.

鈭�abf(x)dx=c(b-a).

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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�(tanx+xsec2x)dx.

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.

Explain why the formula for the integral of xkdoes not

apply whenk=-1. What is the integral of x-1?

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 .

鈭�23x+1dx

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

鈭�鈭�32鈥�2xx2+5dx
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