Question

Suppose $y=f(x)$ and $y=g(x)$ are two continuous functions whose graphs intersect at the three points $(0,4),(2,2)$ and $(4,0)$ with $f(x)>g(x)$ for $0<x<2$ and $f(x)<g(x)$ for $2<x<4$. If $\underset{0}{\overset{4}{\int }}[f(x)-g(x)]dx=10$ and $\underset{2}{\overset{4}{\int }}[g(x)-f(x)]dx=5$, the area between two curves for $0<x<2$, is

• A. 5
• B. 10
• C. 15
• D. 20

Answer 1

Answer: (C)

Given $\underset{0}{\overset{4}{\int }}f(x)dx-\underset{0}{\overset{4}{\int }}g(x)dx=10$
$\left(A_1+A_3+A_4\right)-\left(A_2+A_3+A_4\right)=10$
$A_1-A_2=10\text{....(1) }$
$\text{ again }\underset{2}{\overset{4}{\int }}g(x)dx-\underset{2}{\overset{4}{\int }}f(x)dx=5$
$\left(A_2+A_4\right)-A_4=5$
$A_2=5$ ....(2)
$\therefore (1)+(2)$
$A_1=15$