Integrals and the Fundamental Theorem — AP Calculus AB Calc AB Study Guide

For: AP Calculus AB candidates sitting AP Calculus AB.

Covers: Antiderivatives and indefinite integrals, left/right/midpoint Riemann sums, both parts of the Fundamental Theorem of Calculus, substitution method for integration, and average value of a function over an interval.

You should already know: Strong precalculus (functions, trig, algebra).

A note on the practice questions: All worked questions in the "Practice Questions" section below are original problems written by us in the AP Calculus AB style for educational use. They are not reproductions of past College Board papers and may differ in wording, numerical values, or context. Use them to practise the technique; cross-check with official College Board mark schemes for grading conventions.

1. What Is Integrals and the Fundamental Theorem?

Integral calculus is the inverse of differential calculus, focused on calculating accumulated change, areas under curves, and total quantities from rate functions. The Fundamental Theorem of Calculus (FTC) is the unifying rule that explicitly connects differentiation and integration, eliminating the need for laborious area estimates for most continuous functions. Per the AP Calculus AB CED, this topic accounts for 10-15% of your total exam score, appearing on both multiple choice and free response sections.

2. Antiderivative and indefinite integral

A function $F(x)$ is an antiderivative of $f(x)$ on an interval if $F^{\prime }(x)=f(x)$ for all $x$ in that interval. Since the derivative of any constant is zero, every function has infinitely many antiderivatives that differ by a constant value $C$, called the constant of integration.

The indefinite integral denotes the full family of antiderivatives of $f(x)$, written as: $$\int f(x)dx=F(x)+C$$

Core integration rules (directly inverse of differentiation rules) include:

1. Power rule: $\int x^{n}dx=\frac{x^{n+1}}{n+1}+C$ for $n\ne -1$
2. Trig rules: $\int \sin xdx=-\cos x+C$, $\int \cos xdx=\sin x+C$
3. Constant multiple rule: $\int kf(x)dx=k\int f(x)dx$
4. Sum/difference rule: $\int [f(x)\pm g(x)]dx=\int f(x)dx\pm \int g(x)dx$

Worked Example

Find $\int (3x^2+2\cos x-5)dx$:

1. Integrate term by term:

• $\int 3x^{2}dx=3\cdot \frac{x^3}{3}=x^3$
• $\int 2\cos xdx=2\sin x$
• $\int -5dx=-5x$

2. Add the constant of integration: Final answer = $x^3+2\sin x-5x+C$

Exam tip: You will lose 1 point per missing $+C$ on indefinite integral free response questions, so write it immediately after computing the antiderivative to avoid forgetting.

3. Riemann sums (left, right, midpoint)

Before the FTC allowed exact integral calculations, Riemann sums were used to estimate the area under a curve $f(x)$ over an interval $[a,b]$, which is the definition of the definite integral ${\int }_a^{b}f(x)dx$.

To calculate a Riemann sum:

1. Divide $[a,b]$ into $n$ equal-width subintervals, with width $\Delta x=\frac{b-a}{n}$
2. Define the $i$-th endpoint as $x_i=a+i\Delta x$, so $x_0=a$ and $x_n=b$
3. Multiply the width of each subinterval by the height of $f(x)$ at a sample point in the subinterval, then sum all products.

Three common Riemann sum types:

• Left Riemann sum: Uses the left endpoint of each subinterval for height: $L_n=\Delta x{\sum }_{i=0}^{n-1}f(x_i)$
• Right Riemann sum: Uses the right endpoint of each subinterval for height: $R_n=\Delta x{\sum }_{i=1}^{n}f(x_i)$
• Midpoint Riemann sum: Uses the midpoint of each subinterval for height: $M_n=\Delta x{\sum }_{i=0}^{n-1}f\left(\frac{x_i+x_{i+1}}{2}\right)$

Worked Example

Estimate ${\int }_0^4{x}^{2}dx$ using $n=2$ subintervals for all three sum types:

• $\Delta x=\frac{4-0}{2}=2$, subintervals = $[0,2],[2,4]$
• Left sum: $2\cdot [f(0)+f(2)]=2\cdot (0+4)=8$ (underestimate, since $x^2$ is increasing)
• Right sum: $2\cdot [f(2)+f(4)]=2\cdot (4+16)=40$ (overestimate, since $x^2$ is increasing)
• Midpoint sum: $2\cdot [f(1)+f(3)]=2\cdot (1+9)=20$ (close to the exact value of $\frac{64}{3}\approx 21.33$)

4. FTC parts 1 and 2

The Fundamental Theorem of Calculus has two complementary parts, both tested heavily on the AP Calculus AB exam.

FTC Part 1 (Derivative of an Integral)

If $f$ is continuous on $[a,b]$, then the function $F(x)={\int }_a^{x}f(t)dt$ is differentiable on $(a,b)$, and: $$F^{\prime }(x)=f(x)$$ This means the derivative of an integral with a constant lower bound and variable upper bound is simply the integrand evaluated at the upper bound. For upper bounds that are functions of $x$, apply the chain rule: $$\frac{d}{dx}{\int }_a^{g(x)}f(t)dt=f(g(x))\cdot g^{\prime }(x)$$

Worked Example

Find $\frac{d}{dx}{\int }_2^{x^3}\cos (t^2)dt$: Apply FTC Part 1 and chain rule: $f(g(x))=\cos ((x^3{)}^2)=\cos (x^6)$, $g^{\prime }(x)=3x^2$, so final answer = $3x^2\cos (x^6)$

FTC Part 2 (Evaluate Definite Integrals)

If $f$ is continuous on $[a,b]$, and $F$ is any antiderivative of $f$, then: $${\int }_a^{b}f(x)dx=F(b)-F(a)$$ This rule lets you compute exact definite integrals without Riemann sums, by evaluating the difference of the antiderivative at the interval bounds.

Worked Example

Compute the exact value of ${\int }_0^4{x}^{2}dx$: Antiderivative of $x^2$ is $\frac{x^3}{3}$, so $F(4)-F(0)=\frac{4^3}{3}-0=\frac{64}{3}\approx 21.33$, matching our earlier midpoint sum estimate.

5. Substitution method

The substitution method (also called u-substitution) is the inverse of the chain rule for differentiation, used to integrate composite functions of the form $f(g(x))\cdot g^{\prime }(x)$.

Step-by-step substitution process:

1. Identify the inner composite function $g(x)$ whose derivative is present (or a constant multiple of its derivative is present) in the integrand
2. Set $u=g(x)$, compute $du=g^{\prime }(x)dx$, and rearrange to solve for $dx$ if needed
3. Rewrite the entire integral in terms of $u$, eliminating all $x$ terms
4. Integrate with respect to $u$
5. For indefinite integrals: substitute $g(x)$ back in for $u$ and add $C$
6. For definite integrals: calculate new $u$-bounds using $u=g(x)$ at $x=a$ and $x=b$, then evaluate the antiderivative at the $u$-bounds (no need to substitute back to $x$)

Worked Example 1 (Indefinite Integral)

Find $\int 2x\cos (x^2)dx$:

• Let $u=x^2$, so $du=2xdx$
• Rewrite integral: $\int \cos (u)du=\sin (u)+C$
• Substitute back $u=x^2$: Final answer = $\sin (x^2)+C$

Worked Example 2 (Definite Integral)

Evaluate ${\int }_0^{\sqrt{\pi }}2x\cos (x^2)dx$:

• New $u$-bounds: $x=0\to u=0$, $x=\sqrt{\pi }\to u=\pi$
• Rewrite integral: ${\int }_0^{\pi }\cos (u)du=\sin (u){|}_0^{\pi }=0-0=0$

6. Average value of a function

The average value of a discrete set of points is the sum of the points divided by the number of points. For a continuous function $f(x)$ over $[a,b]$, the average value is the total accumulated value (the definite integral) divided by the length of the interval: $$f_{avg}=\frac{1}{b-a}{\int }_a^{b}f(x)dx$$

The Mean Value Theorem for Integrals extends this rule: if $f$ is continuous on $[a,b]$, there exists at least one point $c\in [a,b]$ where $f(c)=f_{avg}$, meaning the function hits its average value at least once on the interval.

Worked Example

Find the average value of $f(x)=3x^2$ on $[0,2]$, and find the point $c$ where $f(c)=f_{avg}$:

1. Calculate average value: $$f_{avg}=\frac{1}{2-0}{\int }_0^{2}3x^{2}dx=\frac{1}{2}\cdot [x^3{]}_0^2=\frac{1}{2}\cdot 8=4$$
2. Solve for $c$: $3c^2=4\to c=\sqrt{\frac{4}{3}}\approx 1.15$, which lies in $[0,2]$ as required by the Mean Value Theorem for Integrals.

Exam tip: Do not confuse average value with average rate of change, which is $\frac{f(b)-f(a)}{b-a}$. Average value describes the average height of the function, while average rate of change describes the average slope of the function.

7. Common Pitfalls (and how to avoid them)

• Pitfall 1: Forgetting the constant of integration $C$ on indefinite integrals Why students do it: Focus on computing the antiderivative and skip the $C$, since derivatives eliminate constants. Correct move: Write $+C$ immediately after computing the antiderivative, even before simplifying, to avoid missing it. Examiners deduct 1 point per missing $C$ on FRQs.
• Pitfall 2: Mixing up indices for left/right Riemann sums Why students do it: Memorize sum formulas without mapping to subinterval endpoints. Correct move: Explicitly list subinterval endpoints first before writing the sum. Left sums use the first $n$ endpoints ($x_0$ to $x_{n-1}$), right sums use the last $n$ endpoints ($x_1$ to $x_n$).
• Pitfall 3: Applying FTC Part 1 to integrals with variable lower bounds Why students do it: Memorize FTC Part 1 only for constant lower bounds. Correct move: Flip the integral and add a negative sign to move the variable to the upper bound: ${\int }_x^{a}f(t)dt=-{\int }_a^{x}f(t)dt$, so the derivative is $-f(x)$.
• Pitfall 4: Using original $x$-bounds for $u$-substitution on definite integrals Why students do it: Forget that $u$ is a different variable with different bounds. Correct move: Calculate new $u$-bounds immediately after defining $u=g(x)$, and write them next to the integral sign before integrating.
• Pitfall 5: Confusing average value and average rate of change Why students do it: Both formulas include the $\frac{1}{b-a}$ factor. Correct move: Remember average value includes an integral of $f(x)$, while average rate of change uses a difference of $f(x)$ values. If the question asks for average height of the curve, it is asking for average value.

8. Practice Questions (AP Calculus AB Style)

Question 1 (No calculator, 3 points)

a) Compute the indefinite integral $\int (4x^3-2\sin x+\sqrt{x})dx$ (1 point) b) Evaluate the definite integral ${\int }_1^4(4x^3-2\sin x+\sqrt{x})dx$ (2 points)

Solution

a) Integrate term by term: $$\int 4x^{3}dx=x^4,\int -2\sin xdx=2\cos x,\int \sqrt{x}dx=\frac{2}{3}{x}^{3/2}$$ Final answer: $x^4+2\cos x+\frac{2}{3}{x}^{3/2}+C$ (1 point awarded for correct antiderivative with $+C$) b) Use FTC Part 2: Evaluate at $x=4$: $4^4+2\cos (4)+\frac{2}{3}(4{)}^{3/2}=256+2\cos 4+\frac{16}{3}$ Evaluate at $x=1$: $1^4+2\cos (1)+\frac{2}{3}(1{)}^{3/2}=1+2\cos 1+\frac{2}{3}$ Subtract: $\frac{799}{3}+2(\cos 4-\cos 1)\approx 257.28$ (2 points: 1 for correct antiderivative, 1 for correct evaluation)

Question 2 (Calculator allowed, 4 points)

The function $v(t)=10t-t^2$ gives the velocity of a particle in m/s, for $t\in [0,10]$ seconds. a) Use a midpoint Riemann sum with $n=5$ subintervals to estimate the total distance traveled by the particle over $[0,10]$ (2 points) b) Find the exact total distance traveled using FTC Part 2 (2 points)

Solution

a) $\Delta x=\frac{10-0}{5}=2$, midpoints of subintervals = $1,3,5,7,9$ Midpoint sum = $2\cdot [v(1)+v(3)+v(5)+v(7)+v(9)]=2\cdot (9+21+25+21+9)=170$ m (2 points: 1 for correct $\Delta x$ and midpoints, 1 for correct final answer) b) $v(t)\ge 0$ on $[0,10]$, so distance = ${\int }_0^{10}(10t-t^2)dt$ Antiderivative: $5t^2-\frac{t^3}{3}$, evaluate: $5(10{)}^2-\frac{10^3}{3}=\frac{500}{3}\approx 166.67$ m (2 points: 1 for correct integral setup, 1 for correct exact answer)

Question 3 (No calculator, 3 points)

a) Find $\frac{d}{dx}{\int }_{\sin x}^3{e}^{t^2}dt$ (2 points) b) Find the average value of $f(x)=e^{2x}$ on $[0,\ln 3]$ (1 point)

Solution

a) Flip the integral to get a constant lower bound: $-{\int }_3^{\sin x}{e}^{t^2}dt$ Apply FTC Part 1 and chain rule: $-e^{(\sin x{)}^2}\cdot \cos x=-e^{{\sin }^{2}x}\cos x$ (2 points: 1 for flipping integral and adding negative sign, 1 for correct chain rule application) b) Average value: $\frac{1}{\ln 3}{\int }_0^{\ln 3}{e}^{2x}dx=\frac{1}{\ln 3}\cdot {\left[\frac{1}{2}{e}^{2x}\right]}_0^{\ln 3}=\frac{1}{\ln 3}\cdot \frac{1}{2}(9-1)=\frac{4}{\ln 3}$ (1 point for correct formula and calculation)

9. Quick Reference Cheatsheet

10. What's Next

Mastery of integrals and the Fundamental Theorem is the foundation for all remaining AP Calculus AB topics, including applications of integration (area between curves, volume of solids of revolution, accumulated change problems in motion and population growth) that make up 10-15% of your exam score. You will also use substitution and FTC rules to solve differential equations, another core AB topic, and these skills will be required for nearly every integral question on both the multiple choice and free response sections of the test.

If you struggle with any of the concepts in this guide, or want more practice problems tailored to your weak spots, you can ask Ollie, our AI tutor, for personalized help at any time via the homepage. We also recommend working through official College Board past FRQ questions on integrals to familiarize yourself with examiner grading conventions and common question framing.