3.4 Definite Integrals Part 2

3.4 Definite Integrals (Part 2)



Example:
Given that 3 7 f(x)dx=5 , find the values for each of the following:

(a)  3 7 6f(x)dx (b)  3 7 [3f(x)]dx (c)  7 3 2f(x)dx (d)  3 4 f(x)dx + 4 5 f(x)dx+ 3 7 f(x)dx (e)  3 7 f(x)+7 2 dx


Solution:
(a)  3 7 6f(x)dx =6 3 7 f(x)dx        =6(5)=30 (b)  3 7 [3f(x)]dx= 3 7 3dx 3 7 f(x) dx = [ 3x ] 3 7 5 =[ 3(7)3(3) ]5 =7 (c)  7 3 2f(x)dx = 3 7 2f(x)dx =2 3 7 f(x)dx =2(5) =10 (d)  3 4 f(x)dx + 4 5 f(x)dx+ 3 7 f(x)dx = 3 7 f(x)dx =5 (e)  3 7 f(x)+7 2 dx = 3 7 [ 1 2 f(x)+ 7 2 ]dx = 3 7 1 2 f(x)dx + 3 7 7 2 dx = 1 2 3 7 f(x)dx + [ 7x 2 ] 3 7 = 1 2 (5)+[ 7(7) 2 7(3) 2 ] = 5 2 +14 =16 1 2