The area function, and the two theorems that tie area to antiderivative

What to assume they know

• The definite integral as a single number, and the two-step evaluation, from the previous topic
• The anti derivative and its family, from module m01
• Continuity on a closed interval, from Chapter 5 of Part I
• Reading a graph: ordinates, the region between a curve and the horizontal axis, and what it means for a region to be bounded
• Differentiating a function whose rule is given implicitly rather than by a formula
• The idea of a function of a variable that appears as an end point rather than inside the rule

What they should be able to do

• Describe the region the chapter identifies with a definite integral, naming all four of its boundaries
• Read the chapter's figure correctly, including which of the two shaded regions the area function names
• State what the bracketed concession beside the figure allows, and what it therefore forbids a student from assuming
• Define the area function, and say which letter is fixed and which moves
• State the first theorem and explain what it asserts about a derivative
• State the second theorem and connect it to the evaluation recipe of the previous topic
• Say what the chapter proves and what it only states, and quote the sentence in which it says so
• Differentiate an integral whose upper end is the variable — the first theorem applied
• Compare the Summary's statements of the two theorems against the section's

Where it usually goes wrong

• "The area function is the same as the definite integral." It is a function of where the right-hand edge is put; the definite integral is the one number you get once that edge is fixed at the upper limit. Confusing the two makes Theorem 1 unintelligible, because there is nothing left to differentiate.
• "The lighter and darker strips are both the area function." They are not. The labelled, lighter strip runs from the lower limit to the sliding point and is the area function's value; the darker strip is the rest of the region and is what remains to be swept. Read the figure at high resolution before redrawing it.
• "A definite integral is an area, so it is never negative." The figure assumes a curve above the axis and the chapter says in brackets that the result holds beyond that assumption. Where the curve dips below, the number and the ordinary area part company, and the chapter neither draws that case nor names it.
• "Both theorems are proved in the chapter." Neither is. The chapter says so in one sentence.
• "Theorem 1 and Theorem 2 say the same thing in two ways." Theorem 1 produces an anti derivative out of an area; Theorem 2 evaluates an area through any anti derivative. One creates the bridge and the other crosses it. Students who merge them cannot answer Exercise 7.9 Q10, which needs only the first.
• "Differentiating an integral gives back the integrand — that is Property (I) from module m01." Property (I) is about an indefinite integral. Theorem 1 is about a definite integral with a moving upper end, which is a different object. The two look alike written down and are proved differently — in this edition, one is proved and the other is not proved at all.
• "The area function needs the upper end to be the upper limit." It needs the upper end to range over the interval; fixing it at the upper limit is what turns the function back into the single number.
• "The Summary's statements of the theorems can be used interchangeably with the section's." They differ in three places, one of which widens Theorem 1's conclusion beyond the interval the section states. See section 11.

Questions to check understanding

• Describe the region a stated definite integral measures, naming all four boundaries
• Write down the area function for a stated integrand and a stated lower end
• State the first theorem, and use it to differentiate an integral whose upper end is the variable — the form of Exercise 7.9 Q10
• State the second theorem and derive the two-step evaluation recipe from it
• Say what the chapter proves and what it only states
• Explain why the figure's assumption about the curve does not restrict the theorem
• Identify the difference between the section's and the Summary's statements of the first theorem

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.

• §7.8's heading and §7.8.1's opening (Part II p. 267). The section is named for the connection between the two kinds of integral. Its first subsection names the region: bounded above by the curve, below by the horizontal axis, and on the two sides by the ordinates at the lower and upper limits. Four boundaries.
• Fig 7.1 (Part II p. 267) — the chapter's only figure, and it was read at 300 dots per inch before this brief was written. A curve rising from left to right, labelled with its equation at the top right. Vertical ordinates are drawn at three places along the horizontal axis, marked with the lower limit, an intermediate point, and the upper limit. The strip between the lower limit and the intermediate point is shaded in the lighter of two tones and carries the label of the area function; the strip from the intermediate point to the upper limit is shaded in the darker tone and carries no label. The horizontal axis is drawn as a full line with arrowheads and letters at both ends, and the vertical axis likewise, with the origin marked. The chapter's prose calls the labelled strip the light shaded region, which agrees with what the printed page shows. This distinction is the whole figure. A redraw that shades both strips alike destroys the only thing the picture communicates.
• The bracketed concession (Part II p. 267). Beside the figure the chapter notes in brackets that the curve is being assumed to sit above the axis throughout, and says the statement that follows holds for other functions too. Two things follow. The picture is of a positive function, so the shaded strips are areas in the ordinary sense; and the theorem is not restricted to that case, so a student must not conclude that a definite integral is always positive. The chapter draws no second figure for the general case.
• The area function (§7.8.1, Part II p. 267). Fix the lower end, let the upper end be a point of the closed interval, and read the resulting number as a function of that point. The chapter numbers the defining equation and names the function. This is the one genuinely new construction in the module: the variable has moved from inside the integrand to the end of the integral, and everything that follows depends on noticing that.
• The backward reference in the same sentence (§7.8.1, Part II p. 267). The subsection opens by saying the definite integral has already been defined as the area of that region. Where? §7.1 (Part II p. 226) raises finding an area as one of the two problems that motivate the subject; §7.7 (Part II p. 267) defines the definite integral either as a limiting sum or as a difference of anti derivative values, and says nothing about area. So the sentence points back at a definition the chapter has not printed. See section 5 and the note below; this is a reading, and it is flagged as one.
• Theorem 1 (§7.8.2, Part II p. 267). Take an integrand continuous throughout the closed interval; then the area function's derivative is that integrand, at every point of the interval. One sentence, stated and not proved. Verified as the content of section 6: it says the area function is an anti derivative of the integrand — the first time in the chapter that an area and an anti derivative are the same object.
• Theorem 2 (§7.8.3, Part II p. 268). Take an integrand continuous throughout the closed interval and any anti derivative of it; then the definite integral is that anti derivative's value at the upper end less its value at the lower end. Stated and not proved. This is the theorem the previous topic uses on every exercise, and this topic is where it is actually stated. See the note below about how the two theorem statements are worded.
• The sentence about proofs (Part II p. 267, between §7.8.1 and §7.8.2). The chapter says the two theorems are only stated, their proofs being beyond this book's scope. Quote it. It is the honest thing on the page and it matters for section 9: the entire bridge between area and anti derivative rests, in this edition, on two unproved statements.
• Exercise 7.9, question 10 (Part II p. 273). A function is defined as an integral from zero to the variable, and the item asks for its derivative, offering four options. Verified by working added here: this is Theorem 1 applied directly, so the derivative is the integrand evaluated at the variable, which is the second option. It is the only item in the whole chapter that exercises Theorem 1, and it is printed in the substitution exercise two sections later, where a student practising substitution will not be expecting it. Section 10 exists to point at it.
• The Summary (Part II pp. 290–291). Three bullets carry this topic: the region and the area function, then the two theorems under their own headings. Read off the page images. The five-part comparison against the section is section 11 and the differences are set out in the notes below.

Figures to have open

• A faithful redraw of Fig 7.1 (Part II p. 267) for section 2, and reused in sections 1, 4 and 6. It is the chapter's own and the only figure in the chapter. The redraw must keep: a curve rising left to right, labelled with its equation; three ordinates, at the lower limit, the sliding point and the upper limit; two distinct shading tones, with the lighter strip lying between the lower limit and the sliding point and carrying the area function's label; both axes drawn full with arrowheads and the origin marked. Read the original at 300 dots per inch before redrawing; the two tones are close at 100 dots per inch.
• A movement for section 4 in which the sliding ordinate moves and a second curve is traced beneath it, one point at a time. Not in the book; the chapter draws only the static figure.
• A two-column comparison of the two theorem statements against the Summary's for section 11. The content is the chapter's own, from Part II pp. 267–268 and Part II p. 291; the alignment is added here. Build it with the repo's DataTable component.
• No figure is needed for sections 3, 5, 7, 8, 9 or 10, and none is available: this is the chapter's only figure and this topic already uses it.

Where this sits in the book

• NCERT Class 12 Mathematics, Chapter 7 "Integrals", §7.8 Fundamental Theorem of Calculus and §7.8.1 Area function, with Fig 7.1, Part II p. 267
• §7.8.2, headed for the first fundamental theorem, and its Theorem 1, Part II p. 267
• §7.8.3, headed for the second, and its Theorem 2, Part II p. 268
• Exercise 7.9, question 10, Part II p. 273
• Summary, the area-function bullet and the two theorem bullets, Part II pp. 290–291
• §7.1 Introduction, the area problem and the Fundamental Theorem named, Part II pp. 225–226