Choosing vertical or horizontal strips, and integrating in the matching variable

What to assume they know

• The strip construction of the previous topic: a region cut into parallel slices, one slice written as height times width, the total written as an integral
• Rearranging a curve's equation to make either variable the subject
• Recognising when a rearrangement produces two values rather than one, and what that means for a graph
• The standard forms of a circle, an ellipse and a parabola as printed equations
• Definite integration of a power of the variable, from the previous chapter
• The four quadrants, and which pair of axes bounds each

What they should be able to do

• State both of the chapter's area formulas and say which strip direction each one belongs to
• Explain what swaps when the strip turns: which variable supplies the thickness, which supplies the length, and which two lines become the limits
• Identify the real constraint on the choice — the strip's length has to be readable from a single curve as one value
• Set up the same area both ways for a region where either works, and show that the two integrals agree
• Read a region's printed description and decide from it which variable the integral wants
• Set up the horizontal-strip integral for a region bounded by a curve, the vertical axis and a horizontal line
• Predict the answer a student gets by integrating in the wrong variable, and recognise it among a printed set of options
• Check a split of a rectangle into two complementary regions by adding the two areas
• Notice that the chapter writes the second curve with one letter in the body and a different one in its Summary, and treat both as the same object

Where it usually goes wrong

• "Vertical strips are the normal way and horizontal strips are a trick." They are two statements of one construction, printed on the same page with equal standing. The chapter reinforces the wrong impression by demonstrating horizontal strips only as an afterthought on regions it had already done the other way; the exercise that actually needs them arrives with no worked example behind it.
• "Pick whichever strip looks nicer." Looks are the last criterion. The first is whether the strip's length can be read off a single curve as one value everywhere in the region. When only one direction satisfies that, there is no choice to make.
• "You can integrate in whichever variable you like and adjust at the end." The limits belong to the variable. Change the variable and you must change the limits with it and re-read the boundaries, which is three changes, not one. Nothing can be adjusted afterwards.
• "The vertical axis is a boundary like any other." When a region is described as bounded by the vertical axis, that phrase is telling you the strip starts there — it is a signal about the set-up, not a decoration on the description. In Exercise 8.1 question 4 it is the whole clue.
• "The area under a curve and the area beside it are the same region." They are complementary pieces of a rectangle, and they are equal only by accident. In the chapter's own question 4 they are nine quarters and nine halves — one is exactly twice the other.
• "If the answer is one of the printed options it must be right." Two of the chapter's four multiple-choice items carry a distractor that is exactly the answer produced by one specific, common error. Finding your answer on the page is not a check.
• "Rearranging the curve for the other variable is always possible." It is always possible to try; what comes back may be a pair of values rather than one, and then the region has to be cut before the strip has a length. The chapter meets this on Part II p. 294 and disposes of it in a single clause about the quarter it is working in.
• "Two different letters mean two different functions." The body and the Summary write the sideways curve with different letters. Same object.

Questions to check understanding

• Given a region and a chosen strip direction, write the correct integral with its limits, without evaluating it
• State the condition a curve must satisfy before it can supply the strip's length in a chosen direction
• Set up one area both ways and show the two integrals give the same value
• Choose the correct area of a region bounded by a curve, the vertical axis and a horizontal line, from four options — the form of Exercise 8.1 question 4
• Given a wrong answer to such an item, identify the error that produced it
• Add a region and its complement and check the total against the bounding rectangle
• Rewrite a printed curve equation to make the other variable the subject, and say which sign of the root the named region needs

Examples worth working on the board

Values marked verified are worked out here from the chapter's own printed data; neither answers file was opened, and this chapter prints no answers to its exercises.

• The two formulas, printed one above the other (Part II p. 293), with a three-line sentence between them and nothing else. The first runs along the horizontal axis between two vertical lines; the second runs up the vertical axis between two horizontal lines, with the curve written as the first coordinate in terms of the second. The chapter states them, points at a figure for each, and moves straight on to a worked example. It never gives a rule for choosing, and never poses a problem where the choice is the difficulty. That absence is this topic.
• Fig 8.1 against Fig 8.2 (Part II pp. 292–293). Read off the printed pages: in Fig 8.1 the region sits on the horizontal axis, its two vertical sides are the ordinates, and the single strip stands upright with its height labelled inside the region and its width labelled below the axis. In Fig 8.2 the region sits against the vertical axis, its floor and ceiling are two horizontal lines, the curve bulges out to the right, and the single strip lies flat with its length labelled inside the region and its thickness labelled outside the vertical axis. The two figures are the same idea photographed from two sides, and showing them together is the fastest way to make the swap visible.
• The circle both ways (Example 1, Part II pp. 294–295). The vertical-strip set-up integrates the curve's height along the horizontal axis from the centre out to the rightmost point; the alternative integrates the curve's horizontal extent up the vertical axis over the matching run. Verified: both produce the same closed-form antiderivative with the roles of the two variables exchanged, and both land on the same value for the whole disc. This is the chapter's own demonstration that the choice is free when the region is symmetric in the two variables — and note it is a demonstration only, because nothing about that region forces either choice.
• The ellipse both ways (Example 2, Part II pp. 295–296). Same structure. The constant that has to be pulled out of the integral is the ratio of the two semi-axes one way up in the vertical-strip version and the other way up in the horizontal-strip version, and the surviving integral is the corresponding circle integral in each case. Worth one beat: the two constants are reciprocals and the two surviving integrals differ, and the product comes out the same. Note the typesetting slip on Part II p. 296 recorded below before this display is shown.
• Exercise 8.1 question 4 (Part II p. 296). A parabola whose squared variable is the second one, together with the vertical axis and a horizontal line at height three; four numerical options are offered. This is the only item in the chapter where the horizontal strip is not an alternative but the answer. Verified: rearranging gives the horizontal extent as a quarter of the square of the height, and integrating that up the vertical axis from zero to three gives nine quarters, which is the second option.
• The distractor in that item, and it is not an accident. Verified: the region under the same arc measured the other way — vertical strips along the horizontal axis from zero out to nine quarters — comes to nine halves, and nine halves is the fourth printed option. A student who reaches for the familiar set-up without reading which boundaries are named will find their answer on the page and mark it with confidence. Build a whole section on this.
• The ten-second check (not in the book). Verified: the two regions together fill the rectangle three high and nine quarters wide, whose area is twenty-seven quarters; and nine quarters plus nine halves is twenty-seven quarters. The chapter offers no such check anywhere. It costs one line and it catches the wrong-variable error every time.
• Exercise 8.1 question 3 (Part II p. 296). A quarter of a disc, described by the circle and two vertical lines with the quadrant named. Verified: the answer is pi, and the vertical-strip integral is the chapter's own Example 1 restricted to one quarter. Use it as the contrast to question 4 — same chapter, same page, opposite choice — and note the printed oddity about the second line, recorded under Notes.
• The two Summary bullets (Part II p. 298). One for each strip direction. The second bullet writes the curve with a different letter from the one the body uses on Part II p. 293. Verified by reading both pages at 300 dots per inch: the object is the same — the first coordinate written as a function of the second — and only the letter changes. Say so; a student who believes two different constructions are being described will look for a difference that is not there.
• The two "why" prompts (Part II pp. 295 and 296). Each alternative derivation carries a bracketed question mark at the step where the curve is rearranged for the other variable. The chapter never answers either. Verified: in both cases the answer is the same one sentence — the rearrangement is legitimate because the quarter being measured lies where both coordinates are positive, so the square root takes its positive value. Answer them; they are the chapter's only invitation to the reader in this whole topic.

Figures to have open

• Redraws of Fig 8.1 and Fig 8.2 (Part II pp. 292–293) drawn to the same scale and placed side by side, with both strips, both sets of labels, and both pairs of boundary lines. These are the chapter's own diagrams; the pairing is added here, and it carries sections 1 and 2.
• A rotating-strip movement for section 2. The same strip, the same region, one right angle of rotation, labels moving with it. Not in the book.
• A region with a horizontal cut meeting a curve twice for section 4. An added device; the chapter never draws a cut and never illustrates the failure case.
• The Exercise 8.1 question 4 region drawn for section 8, with the parabola, the vertical axis, the horizontal line at height three, the meeting point at nine quarters marked, and one flat strip across it. The chapter prints no figure for any exercise item, so this is entirely added here, and it is the most important new drawing in this brief.
• The same region and its complement inside their bounding rectangle for section 10, with the three areas labelled. Not in the book.

Where this sits in the book

• NCERT Class 12 Mathematics, Chapter 8 "Application of Integrals", §8.2, the vertical-strip construction and Fig 8.1, Part II p. 292
• The horizontal-strip formula and Fig 8.2, Part II p. 293
• Example 1 with Fig 8.5, Part II p. 294, and its alternative with Fig 8.6, Part II p. 295
• Example 2 with Fig 8.7, Part II p. 295, and its alternative with Fig 8.8, Part II p. 296
• Exercise 8.1, questions 3 and 4, Part II p. 296; Summary, both bullets, Part II p. 298