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Calculus - Integration

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    Study Guide Calculus – Integration 1. Distance, Velocity, and Acceleration Many real - world calculus problems involve motion , such as objects moving through space over time. In these problems, distance , velocity , and acceleration are all functions of time, and calculus helps us understand how they are connected. How These Ideas Are Related • The derivative of distance gives instantaneous velocity . • The derivative of velocity gives instantaneous acceleration . • The indefinite integral of acceleration gives velocity. • The indefinite integral of velocity gives distance. In short, derivatives describe how motion changes , and integrals help us recover motion from those changes . Motion Under Gravity For objects moving straight up or down near Earth’s surface, gravity causes a constant acceleration of: The negative sign means the acceleration is directed downward . As time increases, the velocity decreases if the object is moving upward, or becomes more negative if the object is falling downward. Finding Velocity from Acceleration Since velocity is the integral of acceleration:

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    Study Guide Substitute ( a(t) = - 32 ) : To find the constant ( C 1 ), use the initial velocity . If the initial velocity at ( t = 0 ) is ( v 0 ) , then: So the velocity function becomes: Finding Distance from Velocity Distance is the integral of velocity: Substitute ( v(t) = - 32t + v 0 ) : To find ( C 2 ) , use the initial position ( s 0 ) : So the distance function is:

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    Study Guide Example 1 : Time to Hit the Ground A ball is thrown downward from a height of 512 feet with an initial velocity of 64 ft/sec. How long does it take to reach the ground? Step 1: Identify the Given Information • ( a(t) = - 32 ) ft/sec² • ( v 0 = - 64 ) ft/sec (negative because it’s downward) • ( s 0 = 512 ) ft Step 2: Write the Distance Function Step 3: Set Distance Equal to Zero The ball reaches the ground when ( s(t) = 0 ) : Factor: Step 4: Solve Since time cannot be negative:

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    Study Guide Final Answer The ball reaches the ground 4 seconds after it is thrown. Example 2 : Velocity When the Ball Hits the Ground What is the velocity of the ball when it hits the ground? Using the velocity function: Substitute ( t = 4 ) : Final Answer The ball hits the ground with a velocity of – 192 ft/sec . The negative sign indicates the ball is moving downward , meaning the distance from the ground is decreasing as time increases. Example 3 : Missile Launch Problem A missile accelerates upward at 4 m/sec² from rest in a silo 35 meters below ground level. How high above the ground is it after 6 seconds? Step 1: Identify the Given Information • ( a(t) = 4 ) m/sec² • ( v 0 = 0 ) m/sec (starts from rest) • ( s 0 = - 35 ) m (below ground level) Step 2: Find the Velocity Function

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    Study Guide Step 3: Find the Distance Function Step 4: Evaluate at ( t = 6 ) Final Answer After 6 seconds, the missile is 109 meters above the ground . Key Takeaways • Velocity is the derivative of distance. • Acceleration is the derivative of velocity. • Integrals reverse derivatives and help build motion equations. • Initial conditions determine constants of integration. • Signs matter — they tell you the direction of motion. 2. Definite Integrals How Definite Integrals Are Defined To understand the idea of a definite integral, we start with a function ( f(x) ) that is continuous on a closed interval ([a, b]).

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    Study Guide Step 1: Divide the Interval First, the interval ([a, b]) is divided into ( n ) smaller subintervals. These subintervals are often chosen to have equal length, called ( Δ x ) , although this is not strictly required. Step 2: Choose Sample Points From each subinterval, we select a point ( x i ) . For each of these points, we evaluate the function to get ( f( x i ) ) . Step 3: Form Products Next, we multiply each function value by the width of its subinterval: Each of these products represents a small contribution to the total value we are trying to approximate. Step 4: Add Everything Together Finally, we add all these products to form a Riemann sum : or, using summation notation, Understanding the Riemann Sum A Riemann sum can be thought of as a “sum of ( n ) products.” Its value depends on how the function behaves over the interval ([a, b]): • If ( f(x) > 0 ) on ([a, b]), the Riemann sum is positive . • If ( f(x) < 0 ) on ([a, b]), the Riemann sum is negative . • In some cases, positive and negative values may balance out, resulting in a sum of zero .

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    Study Guide Riemann sums are important because they provide the foundation for defining the definite integral. As we increase the number of subintervals and make them very small, the Riemann sum approaches the exact value of the definite integral. Example 1 : Approximating Area Using a Riemann Sum Let’s work through an example to see how a Riemann sum is calculated step by step. Figure 1 A Riemann sum with four subintervals. Problem Evaluate the Riemann sum for using four subintervals of equal length , where each ( x i ) is chosen as the right endpoint of its subinterval.

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    Study Guide Step 1: Find the Width of Each Subinterval Since the interval ([1,3]) is divided into 4 equal parts, the width of each subinterval is Step 2: Identify the Right Endpoints Starting at ( x = 1 ) and moving right by ( Δ x = 1/2 ) , the right endpoints are: Step 3: Set Up the Riemann Sum The Riemann sum for four subintervals is written as: Substituting the values: Step 4: Evaluate the Function Values Since ( f(x) = x 2 ) : Add them:

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    Study Guide Now multiply by ( Δ x = 1/2 ) : What Happens as We Use More Subintervals? As the number of subintervals increases: • Each subinterval becomes narrower • The approximation becomes more accurate Mathematically, this means: • ( n → ∞ ) • ( Δ x → 0 ) If the Riemann sum approaches a limit as this happens, that limit defines the definite integral . Definition of the Definite Integral If the limit exists, the definite integral of ( f(x) ) on ([a,b]) is defined as: Important Terminology • Integrand : the function being integrated, ( f(x) ) • Variable of integration : the variable ( x ) • Limits of integration : o ( a ) : lower limit o ( b ) : upper limit Even though the symbol ( ∫ ) looks the same as in indefinite integrals, remember: • A definite integral gives a single real number • An indefinite integral represents a family of functions

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    Study Guide When Does a Definite Integral Exist? The existence of the limit of a Riemann sum is very important. • If a function ( f(x) ) is continuous on a closed interval ([a,b]), then: o The definite integral exists o The function is said to be integrable on ([a,b]) Continuity guarantees integrability, but the reverse is not always true. Also, even if a definite integral exists, it may not always be easy to calculate exactly. Properties of Definite Integrals Here are some key rules that help us work with definite integrals:

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