   Chapter 4.3, Problem 52E

Chapter
Section
Textbook Problem

Think About It A function f is defined below. Use geometric formulas to find ∫ 0 12 f ( x )   d x . f ( x ) = { 6 ,                               x > 6 − 1 2 x + 9 ,       x ≤ 6

To determine

To calculate: The value of the integral 012f(x)dx where f(x)={6x>612x+9x6. By the use of geometric formula.

Explanation

Given: The provided integral is,

012f(x)dx

And f(x)={6x>612x+9x6

Formula used: The area of a trapezoid with an altitude of h and parallel bases of lengths a and b which we know that,

(area of trapezoid)=12h(a+b)

And, the area of a rectangle of height a and width b we know that,

area of rectangle=a×b

Calculation: The provided function is a straight line up to x=6 and constant for x>6.

For sketch the graph of the provided function draws a straight line y=6 for x>6 and for draw the line y=x2+9 substitute value x=5 and x=6 in f(x) to get,

Put x=5 in f(x)=x2+9 to get,

f(5)=52+9=132

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