Calculus II: Definite Integrals, Riemann Sums, and Applications – Midterm #1 Review

Definite Integrals and Riemann Sums

Introduction to Definite Integrals

The definite integral is a fundamental concept in calculus, representing the signed area under a curve between two points. It is closely related to the concept of Riemann sums, which approximate the area under a curve by summing the areas of rectangles.

• Definite Integral: The integral of a function f(x) from a to b is denoted as $\int_a^b f(x) dx$.

• Riemann Sum: Approximates the area under f(x) by dividing the interval [a, b] into n subintervals and summing the areas of rectangles.

• General Riemann Sum Formula:

$\int_a^b f(x) dx = \lim_{n \to \infty} \sum_{i=1}^n f(x_i^*) \Delta x$

• Width of Subintervals: $\Delta x = \frac{b-a}{n}$

• Right Endpoint: $x_i = a + i\Delta x$

Summation Formulas

Summation formulas are useful for evaluating Riemann sums and definite integrals.

• $\sum_{i=1}^n i = \frac{n(n+1)}{2}$

• $\sum_{i=1}^n i^2 = \frac{n(n+1)(2n+1)}{6}$

Properties of Definite Integrals

Basic Properties

• $\int_a^b f(x) dx = -\int_b^a f(x) dx$

• $\int_a^a f(x) dx = 0$

• $\int_a^b [cf(x)] dx = c \int_a^b f(x) dx$

The Fundamental Theorem of Calculus (FTC)

Statement and Application

The Fundamental Theorem of Calculus links differentiation and integration, providing a method to evaluate definite integrals.

• FTC Part 1: If $F$ is an antiderivative of $f$, then $\frac{d}{dx} \left[ \int_a^x f(t) dt \right] = f(x)$

• FTC Part 2: If $F$ is an antiderivative of $f$, then $\int_a^b f(x) dx = F(b) - F(a)$

Integration Formulas

Common Integrals

• $\int x^n dx = \frac{x^{n+1}}{n+1} + C$ (for $n \neq -1$)

• $\int \sin(ax) dx = -\frac{1}{a} \cos(ax) + C$

• $\int \cos(ax) dx = \frac{1}{a} \sin(ax) + C$

• $\int \sec^2(ax) dx = \frac{1}{a} \tan(ax) + C$

• $\int \csc^2(ax) dx = -\frac{1}{a} \cot(ax) + C$

• $\int \sec(ax) \tan(ax) dx = \frac{1}{a} \sec(ax) + C$

• $\int \csc(ax) \cot(ax) dx = -\frac{1}{a} \csc(ax) + C$

• $\int \frac{1}{1 + (ax)^2} dx = \frac{1}{a} \arctan(ax) + C$

• $\int \frac{1}{\sqrt{1 - (ax)^2}} dx = \frac{1}{a} \arcsin(ax) + C$

• $\int \frac{1}{x} dx = \ln|x| + C$

Area Between Two Curves

Finding Area

To find the area between two curves, follow these steps:

• Identify the top and bottom functions on the interval [a, b].

• Set up the integral: $\int_a^b [f(x) - g(x)] dx$ where $f(x)$ is the upper curve and $g(x)$ is the lower curve.

• Evaluate the definite integral.

True/False and Conceptual Questions

Key Statements

• If $F$ and $G$ are both antiderivatives of $f$, then $F(x) = G(x) + C$.

• If $F(x) = \int_a^x f(t) dt$, then $F'(x) = f(x)$.

• If $f$ is continuous on [a, b] and $\int_a^b f(x) dx = 0$, then $f(x) = 0$ for all $x$ in [a, b]. (Sometimes false; $f$ could be positive and negative and still integrate to zero.)

• If $|f(x)| \geq |g(x)|$ for all $x$ in [a, b], then $\int_a^b |f(x)| dx \geq \int_a^b |g(x)| dx$.

• If $f$ is continuous and increasing, the right-hand Riemann sum overestimates the definite integral.

• The average value of $f$ on [a, b] is $\text{avg}(f) = \frac{1}{b-a} \int_a^b f(x) dx$.

Evaluating Integrals

Sample Problems

• Evaluate $\int \sec^2(2x) dx$

• Evaluate $\int \sqrt{1-x^2} dx$

• Evaluate $\int \frac{1}{\sqrt{1-x^2}} dx$

• Evaluate $\int \frac{1}{1+x^2} dx$

• Evaluate $\int x^2 e^{x^3} dx$

Applications: Motion and Area

Velocity and Distance

• Given $v(t) = 5t^2 + 3t - 6$, find total distance traveled between $t=1$ and $t=3$ using Riemann sums and the FTC.

• Distance is $\int_{t_1}^{t_2} v(t) dt$.

Area of Triangles and Bounded Regions

• Find the area of a triangle with vertices at (0,1), (1,3), (3,4), and (4,2) using calculus.

• Find the area bounded by $y = -x^2 + 6x$ and $y = x^2 - 2x - 24$.

Tables

Summation Formulas Table

Formula	Description
$\sum_{i=1}^n i$	Sum of first n integers
$\sum_{i=1}^n i^2$	Sum of squares of first n integers

Definite Integral Properties Table

Property	Equation
Reversal of limits	$\int_a^b f(x) dx = -\int_b^a f(x) dx$
Zero width	$\int_a^a f(x) dx = 0$
Constant multiple	$\int_a^b [cf(x)] dx = c \int_a^b f(x) dx$

Additional info:

• Some integration formulas and properties are provided for reference and are standard in calculus courses.

• Practice problems cover both conceptual understanding and computational skills, including applications to motion and area.