Please help me with evaluation

Question

Evaluate 

Options:

1. 10.5
2. 12.5
3. 23
4. 25

Answer 1

From 0 to 3 that function under the integral increases at a positive linear rate from (0 , 2) to (3 , 5), because value within the absolute value sign decreases as x increases, the value under the integral reaching zero when x = 3. From x = 3 to x = 8 the function decreases from (3 , 5) to (8 , 0) at a negative linear rate.

So draw a straight line from ( 0, 2) to ( 3, 5) and then another from ( 3 , 5) to (8 , ). Find the area under the resulting curve and there is your answer, which is 25.

Answer 2

To find this integral, graph it and use geometry to find the area under the function.

The area from 0 to 3 is a "laying down" trapezoid with b₁ = 2 and b₂ = 5 and h = 3 so

A = 1/2(b₁ + b₂)h

A = 1/2(2 + 5)(3)

A = 21/2

The area from 3 to 8 is a triangle with b = 5 and h = 5 so

A = 1/2bh

A = 1/2(5)(5)

A = 25/2

A_total = 21/2 + 25/2 = 46/2 = 23