# Science:Math Exam Resources/Courses/MATH101/April 2005/Question 08 (a)

MATH101 April 2005
Other MATH101 Exams

### Question 08 (a)

An unknown continuous function ${\displaystyle f(x)}$ satisfies ${\displaystyle f(0)=0}$, ${\displaystyle f(4)=8}$, and

${\displaystyle 2x\leq f(x)\leq 6x-x^{2}}$

for ${\displaystyle 0\leq x\leq 4.}$ Also, ${\displaystyle f(x)}$ is nondecreasing on this interval, i.e. it satisfies ${\displaystyle f(c)\leq f(d)}$ for all real numbers ${\displaystyle c}$ and ${\displaystyle d}$ with ${\displaystyle 0\leq c\leq d\leq 4.}$ Let ${\displaystyle I}$ be the value of definite integral ${\displaystyle \int _{0}^{4}f(x)\,dx.}$

(a) Let ${\displaystyle L_{100}}$ be the underestimate for ${\displaystyle I}$ obtained by using a Riemann sum with equal-length subintervals and ${\displaystyle x_{i}^{*}=x_{i-1}}$ (i.e. using the left endpoints of the subintervals), and ${\displaystyle R_{100}}$ be the overestimate obtained by using ${\displaystyle x_{i}^{*}=x_{i}}$ (i.e. the right endpoints). Compute a numerical value for ${\displaystyle R_{100}-L_{100}}$.

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