HardMathematicsHigh School (11–12)Functions and Calculus › Application of IntegralsNumericalA-LevelAPIBJEE Main
A glass roof light is made as a flat panel…
Contributed by StemPy LabTry it on paper first. A hint, the key idea and the worked solution are here when you need them.
A glass roof light is made as a flat panel. In the drawing of the panel, and are in metres. The top edge of the panel is the parabola , and the bottom edge is the V-shaped line . The panel is the region between the V and the parabola, and it is symmetric about the -axis. A steel beam runs along the vertical line . The part of the panel to the right of the beam is tinted. Find the value of at which the parabola meets the right arm of the V. Then find the total area of the panel in square metres. Finally find the area of the tinted part, between the beam and the right edge of the panel, in square metres. Give both areas to two decimal places. The supplier charges by area, so the designer needs good accuracy. Do not round any number until the last step.
Need a reminder? The key ideaShow
Split the region at every crossing of the curves, and integrate (upper lower) on each piece. Use horizontal strips, with width (right left), when the curves are given as or when vertical strips would need splitting.
The worked solution
Have you tried it?
Work it out on paper first. It is fine to get stuck or make a mistake: that is how we learn. When you are ready, go through the solution one step at a time, and check each step against your own.
Step 1. The meeting point. gives , so . For the right arm, . By symmetry the left arm meets at .
Step 2. The right half. For the strip height is . The integral is .
Step 3. The whole panel. .
Step 4. The tinted part. .
Check. The tinted part is a small share of the half panel: .
Answer: the curves meet at m, the panel has area , and the tinted part has area .
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