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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.

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Integral Equation

Solve this integral equation.

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Integral Inequality

An inequality involving integrals of squares of functions.

Integral Sandwich

Stage: 5 Challenge Level: Challenge Level:2 Challenge Level:2

(i) Suppose that $f(0)=0$ and that, for $x\neq 0$, $$0 \leq {f(x)\over x} \leq 1$$ Show that $$-{1\over 2} \leq \int_{-1}^1 f(x)\,dx \leq {1\over 2}$$ (ii) Suppose that $f(0)=0$ and that, for $x\neq 0$, $$0 \leq {f(x)\over x^2} \leq 1$$ Show that $$0 \leq \int_{-1}^1 f(x)\,dx \leq {2\over 3}$$ (iii) Generalize (i) and (ii) to the case where $f(0)=0$ and, for $x\neq 0$, $$0 \leq {f(x)\over x^n} \leq 1$$ where $n$ is a positive integer.