### Area L

Draw the graph of a continuous increasing function in the first quadrant and horizontal and vertical lines through two points. The areas in your sketch lead to a useful formula for finding integrals.

### Integral Sandwich

Generalise this inequality involving integrals.

### Integral Inequality

An inequality involving integrals of squares of functions.

# Integral Equation

##### Stage: 5 Challenge Level:

Thank you for your solutions to Daniel and Ben (no schools given) and to Rajiv from the International School of Seychelles and Shaun from Nottingham High School.

The integral equation is: $$\int_0^x f(t)\,dt = 3f(x)+k,\quad \quad(\star)$$ where $k$ is a constant. Differentiating both sides of $(\star)$ gives $$f(x) = 3f'(x)$$ If there is a solution of $(\star)$ it must be of the form $$f(x) = Ae^{x/3},$$ for some constant $A$. We check to see whether or not this is a solution.

For $f(x)=Ae^{x/3}$ we have $$\int_0^x Ae^{t/3}\,dt = \Big[3Ae^{t/3}\Big]_0^x = 3Ae^{x/3}-3A.$$ Thus $f(x)=Ae^{x/3}$ is a solution if and only if $A=-k/3$. The unique solution is $$f(x) = {-k\over 3} e^{x/3}.$$