One day, during a casual discussion with a student at The Villa, one of the student’s friends asked an interesting question about AP Calculus: Why do area-between-curves problems become confusing as soon as the graphs cross or the orientation changes? She understood integration fairly well but wasn’t sure how to set up the correct integral. The Villa is a residential villa community within the wider Dubailand area, offering easy access to Academic City, Dubai Silicon Oasis, Liwan and other residential developments. Among many questions that these students asked, i managed to answer some of them as I was in a rush. However i also gave them assurance that IBScholars uses visual modelling before integration to teach area and volume problems, and therefore, students will know exactly what their integral represents. Here is how our conversation went.
How do students find the area between two curves?
Dr. Anil Khare: To find the area between two curves, students integrate the difference of the functions over the interval in which the area is bounded.
Why must students find intersection points first?
Dr. Anil Khare: The boundaries of the region may be given by the points of intersection and may show where the upper and lower functions intersect.
When should an integral be split into separate parts?
Dr. Anil Khare: If the mathematical description of the region changes over the interval, then an integral should be split.
How are volume problems different from area problems?
Dr. Anil Khare: Volume problems involve integration to find the cross-sectional area of a three-dimensional solid, not just a planar area.
Why should students sketch the region before integrating?
Dr. Anil Khare: Students should draw the region; as in the diagram, there are boundaries, the direction of the region, and the correct expression to integrate.
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