Unit 4 · Core unit

Integrals and the Fundamental Theorem

The integral arrives twice: as an accumulation built from Riemann sums, and as the reverse of differentiation. The Fundamental Theorem is what joins the two.

What a strong answer looks like

A strong answer distinguishes the definite integral, which is a number, from the indefinite integral, which is a family of functions and needs a constant.

Practice this unit Course overview

Chapter introduction

Understand the idea before memorizing the rule

The integral arrives twice: as an accumulation built from Riemann sums, and as the reverse of differentiation. The Fundamental Theorem is what joins the two.

In Calculus Honors, success means moving among words, symbols, tables, and graphs while preserving the meaning of each quantity. Use the lessons below as connected notes: explain the concept, carry out the procedure, test the result, and interpret it in context.

Detailed study notes

Core lessons, reasoning, and common errors

Lesson 1

Accumulation and the Definite Integral

Concept explanation

A Riemann sum approximates accumulated change with rectangles; the definite integral is the limit as the rectangles narrow. It measures signed area.

How to apply it

Estimate with left, right, and midpoint sums, and say which over- or under-estimates given the direction of the curve.

Common mistake

Treating the integral as area rather than signed area, and losing the sign below the axis.

Guided study move

Estimate the same integral three ways and order the estimates before computing the exact value.

Lesson 2

The Fundamental Theorem, Both Parts

Concept explanation

One part says differentiating an accumulation function returns the integrand; the other says a definite integral can be evaluated from any antiderivative.

How to apply it

Use the first part when the variable is a limit of integration and the second when a number is wanted.

Common mistake

Evaluating an accumulation function first when the question only asks for its derivative.

Guided study move

Differentiate an accumulation function with a variable upper limit, then with a composite upper limit.

Lesson 3

Antiderivatives and Substitution

Concept explanation

Substitution reverses the chain rule. It works when the integrand contains an inner function together with a multiple of its derivative.

How to apply it

Choose u so that du accounts for the leftover factor; for a definite integral, change the limits with the variable.

Common mistake

Omitting the constant of integration, or keeping the original limits after substituting.

Guided study move

Work the same definite integral twice, once converting back to x and once changing the limits.

Formula and visual reference

Connect the symbols to the picture

Chapter synthesis

Ideas to connect

Signed area rather than area
Matching the part of the theorem to the question
Changing limits when substituting

Modeling lab

Transfer the chapter to a new setting

A rate of flow is given as a function of time. Interpret the definite integral over an interval, state its units, and say what the accumulation function represents.

Mastery checklist

• Estimate an integral with a left and a right Riemann sum.
• Evaluate a definite integral using an antiderivative.
• Differentiate an accumulation function.
• Integrate by substitution, definite and indefinite.

Check yourself

• Why does the constant of integration matter for an indefinite integral but not a definite one?
• What does the first part of the theorem say that the second does not?

Modeling drill

A rate of flow is given as a function of time. Interpret the definite integral over an interval, state its units, and say what the accumulation function represents.

Riemann sumDefinite integralSigned areaAntiderivativeFundamental Theorem of CalculusAccumulation functionSubstitution

Unit 3: Applications of Differentiation

Unit 5: Applications of Integration