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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.

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Integral Equation

Solve this integral equation.

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Integral Sandwich

Generalise this inequality involving integrals.

Integral Inequality

Stage: 5 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3
The first part of this question involves an easy evaluation of the integrals and then some simple algebra to compare the results.

In the second part you will get a quadratic inequality in $\lambda$ of the form $$U + 2V\lambda + W\lambda^2 \geq 0.$$ Completing the square leads to a condition for this inequality to hold which gives the required result.