Mathematics

Problem

For the curve

y=kx^{2}+2

with

k>0

, consider the region enclosed by the curve, the

x

-axis, and the vertical lines

x=0

and

x=2

. a. (i) Obtain a definite integral, in terms of

k

, that represents the area of the region. b. (ii) Given that this area equals

14

square units, determine the value of

k

. c. (iii) For the value of

k

obtained in part (ii), find the average height of the curve over the interval

[0,2]