MediumMathematicsHigh School (11–12)Functions and Calculus › Application of IntegralsNumericalA-LevelAPBoard ExamIB
On a map with distances in kilometres, the western shore of a lake is the y-axis…
Contributed by StemPy LabTry it on paper first. A hint, the key idea and the worked solution are here when you need them.
On a map with distances in kilometres, the western shore of a lake is the -axis. The eastern shore is the curve . The lake is bounded in the north by the line and in the south by the line . Find the area of the lake. Use horizontal strips, because the shore is given as in terms of .
Need a reminder? The key ideaShow
Under a curve, between two vertical lines, . Beside the -axis, between two horizontal lines, . An ellipse with semi-axes and has area .
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 strip. At height , the length of a strip is and its thickness is .
Step 2. The integral. .
Step 3. The limits. At the value is . At it is . The difference is .
Answer: the area of the lake is square kilometres.
Watch out. Putting in an integral with would give a different and wrong answer.
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