For , let , where is a positive…

MATHEMATICS · Multi-part · AP · Calculus AB · Applications of Integration · Applications of Integration

Problem

For

0\le x\le 2

, let

h_k(x)=k e^{-x/2}+2

, where

k

is a positive constant. (a) Write, but do not evaluate, an expression for the area of the region bounded above by the graph of

h_k

and below by the

x

-axis on

[0,2]

. (b) Given that this area equals

10

, determine the exact value of

k

. Justify each step of your work. (c) Using the value of

k

found in part (b), compute the average height of

h_k

over

[0,2]

and interpret its meaning in terms of the region’s geometry.