Review question

# Can we solve these simultaneous equations with integrals? Add to your resource collection Remove from your resource collection Add notes to this resource View your notes for this resource

Ref: R6076

## Solution

Given a function $f(x)$, you are told that $\int_0^1 3f(x)\,dx+\int_1^2 2f(x)\,dx=7,$ $\int_0^2 f(x)\,dx+\int_1^2 f(x) \,dx=1.$ It follows that $\int_0^2 f(x) \,dx$ equals

1. $-1$,

2. $0$,

3. $\dfrac{1}{2}$,

4. $2$.

Note that $\int_0^1 af(x)\,dx=a\int_0^1 f(x)\,dx$, where $a$ is a constant, and $\int_0^2 f(x)\,dx=\int_0^1 f(x)\,dx +\int_1^2f(x)\,dx$.

Write $A=\int_0^1 f(x)\,dx$ and $B=\int_1^2f(x)\,dx$. So our two equations become the simultaneous equations $3A+2B=7$ and $A+2B=1.$ Subtracting the second from the first gives $2A=6$, so $A=3$ and $B=-1$.

Now $\int_0^2 f(x) \,dx=A+B=2$.