areas & volumes / kinematics

► syllabus content for the Calculus Topic: SL syllabus (see syllabus section 6.5); HL syllabus (see syllabus sections 6.5).

Whether using integral calculus to compute the area under a curve, area between curves, or the volume of a solid of revolution, I think it's very important for students to have a clear visual understanding of the process. It's often not easy to make sketches for questions involving areas and volumes (especially a 3D sketch for volumes of revolution), but I believe students gain a better understanding by making their own diagrams. In the case of finding an area, a student should sketch a "representative rectangle", and for finding a volume a student should sketch a "representative disc" (or 'washer'). Open the question and solution in the box below for a quick illustration of this.

Areas & Volumes - example of a "representative rectangle" and of a "representative disc"

For both SL and HL, questions involving the displacement $$s\left( t \right)$$, velocity $$v\left( t \right)$$ and acceleration $$a\left( t \right)$$ of an object (i.e. kinematics) will only involve linear motion. The formula for total distance travelled from $${t_1}$$ to $${t_2}$$ is $$\int_{{t_1}}^{{t_2}} {\left| {v\left( t \right)} \right|} \,dt$$.  This formula appears in both of the subject guides for SL and HL and is in the SL formula booklet; however, it is not in the HL formula booklet.  This was probably an oversight.  Using this formula appropriately will definitely help a student answer certain kinematics questions more efficiently - especially questions where a GDC is allowed.  The first exercise set in the teaching materials at the bottom of this page contains kinematics questions (includes worked solutions).

The set of 4 Questions below are on finding areas under or between curves. I will soon be adding a question involving kinematics to this set.

4 questions - ‘accessible’ to ‘discriminating’

accessible SL question

moderate SL / accessible HL question

discriminating SL / moderate HL question

discriminating HL question

Course planning / teaching notes:

There are essentially three important differences between the SL and HL syllabus content with regard to applying integral calculus to finding areas & volumes.

1. In HL, students may be asked to find the area of a region enclosed by a curve and the y-axis whereas SL is only responsible for finding area of a region enclosed by a curve and the x-axis.

2. HL includes finding volumes of solids generated by revolving a curve either about the x-axis, or the y-axis. SL only considers solids generated by a curve revolved about the x-axis.

3. Questions that ask HL students to find the area under a curve, area between two curves, or volume of a solid of revolution will generally be more sophisticated and may demand a higher degree of problem solving compared to those posed to SL students.

♦ teaching materials

EXS_6-6-40v1_SLHL_kinematics
set of 4 exercises (GDC allowed on all) involving displacement, velocity & acceleration of an object moving along a line. Worked solutions are included.

EXS_6-5-25v1_SLHL_areas_volumes
set of 5 exercises on applying integral calculus to find the area under and between curves, and the volume of solids of revolutions about the x-axis; 3 questions with no GDC, and 2 questions with GDC allowed; worked solutions included

WRK_6-5-30v1_SLHL_area_under_curves
worksheet containing three worked examples with notes; and 5 exercises (answers included)

EXS_6-5-35v1_SLHL_def_intgrls_area_volume
Set of 9 exercises (GDC allowed on all) that covers: area under a curve; area between two curves; and volumes of revolution about the x-axis. This set of exercises was designed so that students repeatedly need to consider whether or not use of the GDC is possible and/or appropriate given a question's instructions.

Quiz_HL_integration_v1
content on integration Quiz (HL): integration by substitution; integration by parts; area of a region enclosed by a curve and the x-axis; area of a region enclosed by curves; appropriate and effective use of a GDC (solution key available below). 5 questions - GDC allowed on all questions. All of the questions - except #4 - are suitable for SL students.

Quiz_HL_integration_v1_Sol_Key
worked solutions for above integration Quiz (HL)

EXS_6-5-45v2_SLHL_volumes_revolution
set of 9 exercises on finding volumes of solids of revolution with 6 not allowing a GDC and 3 allowing a GDC; two of the exercises are only for HL, and there is a 'challenge question' at the end; answers included

EXS_6-5-55v1_HL_vol_solids_of_rev1
set of 7 exercises (4 with no GDC; 3 with GDC allowed) on volumes of solids of revolution suitable for HL students (answers included)

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