This is the capstone review for Calculus 2. It ties together integration techniques, applications of integration, differential equations, and infinite series.
Basic integration and u-substitution
Integration by parts
Trigonometric integrals and trig substitution
Partial fractions
Improper integrals
Area between curves
Volumes by disks/washers and shells
Arc length and surface area of revolution
Work and average value
Physics applications (center of mass, fluid force)
Introduction and classification
Separable equations
Linear equations with integrating factors
Sequences, series, and partial sums
Convergence tests (nth-term, p-series, comparison, ratio, root)
Alternating series and absolute/conditional convergence
Power series and intervals of convergence
Taylor and Maclaurin series
Applications of Taylor series
Integration by parts
∫ u d v = u v − ∫ v d u \int u\,dv = uv - \int v\,du ∫ u d v = uv − ∫ v d u Trig substitution patterns
a 2 − x 2 → x = a sin θ a 2 + x 2 → x = a tan θ x 2 − a 2 → x = a sec θ \sqrt{a^2 - x^2} \to x = a\sin\theta \qquad \sqrt{a^2 + x^2} \to x = a\tan\theta \qquad \sqrt{x^2 - a^2} \to x = a\sec\theta a 2 − x 2 → x = a sin θ a 2 + x 2 → x = a tan θ x 2 − a 2 → x = a sec θ Improper integrals: replace the problematic bound with a variable and take the limit
∫ a ∞ f ( x ) d x = lim b → ∞ ∫ a b f ( x ) d x \int_a^{\infty} f(x)\,dx = \lim_{b \to \infty} \int_a^b f(x)\,dx ∫ a ∞ f ( x ) d x = b → ∞ lim ∫ a b f ( x ) d x Area between curves
A = ∫ a b [ f ( x ) − g ( x ) ] d x where f ( x ) ≥ g ( x ) A = \int_a^b \left[f(x) - g(x)\right]dx \quad \text{where } f(x) \geq g(x) A = ∫ a b [ f ( x ) − g ( x ) ] d x where f ( x ) ≥ g ( x ) Volumes of revolution
V disk = π ∫ a b [ R ( x ) ] 2 d x V washer = π ∫ a b [ R outer 2 − R inner 2 ] d x V_{\text{disk}} = \pi\int_a^b [R(x)]^2\,dx \qquad V_{\text{washer}} = \pi\int_a^b \left[R_{\text{outer}}^2 - R_{\text{inner}}^2\right]dx V disk = π ∫ a b [ R ( x ) ] 2 d x V washer = π ∫ a b [ R outer 2 − R inner 2 ] d x V shell = 2 π ∫ a b x h ( x ) d x V_{\text{shell}} = 2\pi\int_a^b x\,h(x)\,dx V shell = 2 π ∫ a b x h ( x ) d x Arc length
L = ∫ a b 1 + [ f ′ ( x ) ] 2 d x L = \int_a^b \sqrt{1 + [f'(x)]^2}\,dx L = ∫ a b 1 + [ f ′ ( x ) ] 2 d x Work by a variable force
W = ∫ a b F ( x ) d x W = \int_a^b F(x)\,dx W = ∫ a b F ( x ) d x Average value
f avg = 1 b − a ∫ a b f ( x ) d x f_{\text{avg}} = \frac{1}{b - a}\int_a^b f(x)\,dx f avg = b − a 1 ∫ a b f ( x ) d x Separable: separate variables and integrate both sides
d y d x = f ( x ) g ( y ) ⇒ ∫ 1 g ( y ) d y = ∫ f ( x ) d x \frac{dy}{dx} = f(x)\,g(y) \quad \Rightarrow \quad \int \frac{1}{g(y)}\,dy = \int f(x)\,dx d x d y = f ( x ) g ( y ) ⇒ ∫ g ( y ) 1 d y = ∫ f ( x ) d x Linear first-order: multiply by the integrating factor
d y d x + P ( x ) y = Q ( x ) μ ( x ) = e ∫ P ( x ) d x \frac{dy}{dx} + P(x)\,y = Q(x) \qquad \mu(x) = e^{\int P(x)\,dx} d x d y + P ( x ) y = Q ( x ) μ ( x ) = e ∫ P ( x ) d x Geometric series
∑ n = 0 ∞ a r n = a 1 − r when ∣ r ∣ < 1 \sum_{n=0}^{\infty} ar^n = \frac{a}{1 - r} \quad \text{when } |r| \lt 1 n = 0 ∑ ∞ a r n = 1 − r a when ∣ r ∣ < 1 p-series
∑ n = 1 ∞ 1 n p converges if p > 1 , diverges if p ≤ 1 \sum_{n=1}^{\infty} \frac{1}{n^p} \quad \text{converges if } p \gt 1, \quad \text{diverges if } p \leq 1 n = 1 ∑ ∞ n p 1 converges if p > 1 , diverges if p ≤ 1 Ratio test
L = lim n → ∞ ∣ a n + 1 a n ∣ converges if L < 1 , diverges if L > 1 L = \lim_{n \to \infty} \left|\frac{a_{n+1}}{a_n}\right| \quad \text{converges if } L \lt 1, \quad \text{diverges if } L \gt 1 L = n → ∞ lim a n a n + 1 converges if L < 1 , diverges if L > 1 Alternating series test: converges if terms decrease and approach zero
Power series centered at a
∑ n = 0 ∞ c n ( x − a ) n converges when ∣ x − a ∣ < R \sum_{n=0}^{\infty} c_n(x - a)^n \quad \text{converges when } |x - a| \lt R n = 0 ∑ ∞ c n ( x − a ) n converges when ∣ x − a ∣ < R Taylor series centered at a
f ( x ) = ∑ n = 0 ∞ f ( n ) ( a ) n ! ( x − a ) n f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(a)}{n!}(x - a)^n f ( x ) = n = 0 ∑ ∞ n ! f ( n ) ( a ) ( x − a ) n Lagrange remainder (error bound)
R n ( x ) = f ( n + 1 ) ( c ) ( n + 1 ) ! ( x − a ) n + 1 R_n(x) = \frac{f^{(n+1)}(c)}{(n+1)!}(x - a)^{n+1} R n ( x ) = ( n + 1 )! f ( n + 1 ) ( c ) ( x − a ) n + 1 Common Maclaurin series
e x = ∑ n = 0 ∞ x n n ! sin x = ∑ n = 0 ∞ ( − 1 ) n x 2 n + 1 ( 2 n + 1 ) ! cos x = ∑ n = 0 ∞ ( − 1 ) n x 2 n ( 2 n ) ! e^x = \sum_{n=0}^{\infty} \frac{x^n}{n!} \qquad \sin x = \sum_{n=0}^{\infty} (-1)^n \frac{x^{2n+1}}{(2n+1)!} \qquad \cos x = \sum_{n=0}^{\infty} (-1)^n \frac{x^{2n}}{(2n)!} e x = n = 0 ∑ ∞ n ! x n sin x = n = 0 ∑ ∞ ( − 1 ) n ( 2 n + 1 )! x 2 n + 1 cos x = n = 0 ∑ ∞ ( − 1 ) n ( 2 n )! x 2 n 1 1 − x = ∑ n = 0 ∞ x n for ∣ x ∣ < 1 \frac{1}{1-x} = \sum_{n=0}^{\infty} x^n \quad \text{for } |x| \lt 1 1 − x 1 = n = 0 ∑ ∞ x n for ∣ x ∣ < 1 1. Integration by parts
∫ x e x d x \int xe^x\,dx ∫ x e x d x u = x , d v = e x d x , d u = d x , v = e x u = x, \quad dv = e^x\,dx, \quad du = dx, \quad v = e^x u = x , d v = e x d x , d u = d x , v = e x ∫ x e x d x = x e x − ∫ e x d x = x e x − e x + C = e x ( x − 1 ) + C \int xe^x\,dx = xe^x - \int e^x\,dx = xe^x - e^x + C = e^x(x - 1) + C ∫ x e x d x = x e x − ∫ e x d x = x e x − e x + C = e x ( x − 1 ) + C 2. Volume by shells
Region between y = x and y = x² from x = 0 to x = 1, rotated around the y-axis.
V = 2 π ∫ 0 1 x ( x − x 2 ) d x = 2 π ∫ 0 1 ( x 2 − x 3 ) d x V = 2\pi\int_0^1 x(x - x^2)\,dx = 2\pi\int_0^1 (x^2 - x^3)\,dx V = 2 π ∫ 0 1 x ( x − x 2 ) d x = 2 π ∫ 0 1 ( x 2 − x 3 ) d x = 2 π [ x 3 3 − x 4 4 ] 0 1 = 2 π ( 1 3 − 1 4 ) = 2 π ⋅ 1 12 = π 6 = 2\pi\left[\frac{x^3}{3} - \frac{x^4}{4}\right]_0^1 = 2\pi\left(\frac{1}{3} - \frac{1}{4}\right) = 2\pi\cdot\frac{1}{12} = \frac{\pi}{6} = 2 π [ 3 x 3 − 4 x 4 ] 0 1 = 2 π ( 3 1 − 4 1 ) = 2 π ⋅ 12 1 = 6 π 3. Separable differential equation
Solve dy/dx = 2xy.
1 y d y = 2 x d x ⇒ ln ∣ y ∣ = x 2 + C ⇒ y = A e x 2 \frac{1}{y}\,dy = 2x\,dx \quad \Rightarrow \quad \ln|y| = x^2 + C \quad \Rightarrow \quad y = Ae^{x^2} y 1 d y = 2 x d x ⇒ ln ∣ y ∣ = x 2 + C ⇒ y = A e x 2 4. Convergence test (ratio test)
Does the series converge?
∑ n = 0 ∞ x n n ! \sum_{n=0}^{\infty} \frac{x^n}{n!} n = 0 ∑ ∞ n ! x n L = lim n → ∞ ∣ x ∣ n + 1 / ( n + 1 ) ! ∣ x ∣ n / n ! = lim n → ∞ ∣ x ∣ n + 1 = 0 < 1 L = \lim_{n \to \infty} \frac{|x|^{n+1}/(n+1)!}{|x|^n/n!} = \lim_{n \to \infty} \frac{|x|}{n+1} = 0 \lt 1 L = n → ∞ lim ∣ x ∣ n / n ! ∣ x ∣ n + 1 / ( n + 1 )! = n → ∞ lim n + 1 ∣ x ∣ = 0 < 1 Converges for all x. This is the series for e to the x.
Calculus 2 skills show up everywhere:
Computing areas, volumes, and accumulated quantities in engineering
Modeling continuous change with differential equations in physics and biology
Approximating complex functions with series in game engines and scientific computing
Analyzing convergence in numerical methods and machine learning
You’ve worked through advanced integration techniques, applications of integration, differential equations, and infinite series. That’s a massive amount of ground. Some of these topics will keep showing up in Calculus 3, differential equations courses, and beyond. The tools you’ve built here, especially series and integration techniques, are genuinely useful in science, engineering, and computing. Well done.
Integration by parts reverses A. the chain rule B. the quotient rule C. the product rule D. the power rule
The shell method formula when rotating around the y-axis is A. $\pi\int [R]^2\,dx$ B. $2\pi\int x\,h(x)\,dx$ C. $\pi\int [R_{\text{out}}^2 - R_{\text{in}}^2]\,dx$ D. $\int f(x)\,dx$
An improper integral converges if A. the integrand is continuous B. it has finite bounds C. the limit exists and is finite D. the function is positive
The average value of $f(x)$ on $[a, b]$ equals A. $f(b) - f(a)$ B. the maximum of f C. $\frac{1}{b-a}\int_a^b f(x)\,dx$ D. the slope at the midpoint
A separable DE $\frac{dy}{dx} = f(x)g(y)$ is solved by A. using an integrating factor B. guessing the solution C. separating variables and integrating both sides D. taking the second derivative
The alternating series test requires the terms $a_n$ to be A. increasing B. constant C. decreasing and approaching zero D. all positive
A power series converges when A. only at the center B. $|x - a| \gt R$ C. $|x - a| \lt R$ D. for all x always
The Maclaurin series for $e^x$ is A. $\sum (-1)^n x^{2n}/(2n)!$ B. $\sum x^n$ C. $\sum (-1)^n x^{2n+1}/(2n+1)!$ D. $\sum x^n/n!$
Taylor series approximate functions with A. trigonometric identities B. polynomials centered at a point C. geometric shapes D. differential equations
The p-series $\sum 1/n^p$ converges when A. $p = 1$ B. $p \leq 1$ C. $p \gt 1$ D. for all $p$
When curves cross in an area problem, you should A. ignore the crossing B. split the integral at the intersection points C. only integrate positive parts D. use the shell method
The integrating factor for $\frac{dy}{dx} + P(x)y = Q(x)$ is A. $P(x)$ B. $Q(x)$ C. $e^{\int P(x)\,dx}$ D. $\int Q(x)\,dx$
For trig substitution with $\sqrt{a^2 - x^2}$, use A. $x = a\tan\theta$ B. $x = a\sec\theta$ C. $x = a\sin\theta$ D. $x = a\cos\theta$
A series converges absolutely if A. the terms alternate B. $\sum |a_n|$ converges C. the nth term is zero D. it converges conditionally
The ratio test is especially useful for series with A. only polynomials B. only constants C. factorials or exponentials D. only alternating signs
The Lagrange remainder bounds A. the radius of convergence B. the error of a Taylor polynomial approximation C. the number of terms needed D. the derivative at the center
The harmonic series $\sum 1/n$ A. converges to 1 B. converges to ln 2 C. diverges D. converges conditionally
The disk method formula for volumes is A. $2\pi\int x\,h(x)\,dx$ B. $\int f(x)\,dx$ C. $\pi\int [R(x)]^2\,dx$ D. $\pi\int [f'(x)]^2\,dx$
The Maclaurin series for $\sin x$ contains only A. even powers B. all powers C. odd powers D. constant terms
After Calculus 2, you should feel confident with A. only memorizing formulas B. only graphing C. integration techniques, applications, DEs, and series D. only abstract proofs
Retry Quiz
