# areas & volumes / kinematics

** Quick links**:

► downloadable teaching materials for areas & volumes

► 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.

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’

__4 questions__-

download: 4_Qs_areas_1_with_answers

#### 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)