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Chapter 4: Q. 80 (page 375)

Use evaluation notation and the Fundamental Theorem of Calculus to prove Theorem 4.26:

鈭�abf(x)dx=鈭�f(x)dxab

Short Answer

Expert verified

Ans:鈭�ab鈥�f(x)dx=[F(x)]ab=鈭�f(x)dxab

Step by step solution

01

Step 1. Given Information: 

鈭�abf(x)dx=鈭�f(x)dxab

02

Step 2. Prove:

ThefundamentaltheoremofCalculusis:鈭�abf(x)dx=limn鈫�鈭�鈭�k=1nFxk-Fxk-1=limn鈫�鈭�F(b)-F(a)=F(b)-F(a)Where,Fisananti-derivativeoffso,鈭�ab鈥�f(x)dx=[F(x)]ab=鈭�f(x)dxabHenceproved.

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