Mathematics

Problem

For a continuous function

f

[0,3]

it is known that

\int_{0}^{3}\bigl(f(x)+k\bigr)\,dx = 12+3k

for every real number

k

. In addition,

\int_{0}^{2} f(x)\,dx = 7.

Using this information, answer the following. a. (i) Find the numerical value of

\displaystyle \int_{0}^{3} f(x)\,dx

. Show your reasoning. b. (ii) Hence calculate

\displaystyle \int_{2}^{3} f(x)\,dx

. Write each step. c. (iii) Determine the value of

k

for which

\displaystyle \int_{0}^{3}\bigl(2f(x)+k\bigr)\,dx = 20

. Give complete working.