6.1 Exploring Accumulations of Change

• Match the Riemann sum type with its defining characteristic:

  Left Riemann Sum ↔️ Uses left endpoint

  Right Riemann Sum ↔️ Uses right endpoint

  Midpoint Riemann Sum ↔️ Uses midpoint

  Trapezoidal Rule ↔️ Uses trapezoids
• The formula for the Left Riemann Sum is \sum_{i = 1}^{n} f(x_{i - 1}) \Delta x
• The Right Riemann Sum uses the right endpoint of each subinterval to determine the height of the rectangle.
• The formula for the Midpoint Riemann Sum is \sum_{i = 1}^{n} f\left(\frac{x_{i - 1} + x_{i}}{2}\right) \Delta x
• What shape is used to approximate the area under a curve in the Trapezoidal Rule?

  Trapezoids
• Approximating $\int_{0}^{2} x^{2} dx$ using the Left Riemann Sum with 4 subintervals gives 2.75.
• Order the Riemann sum types from lowest to highest approximation of $\int_{0}^{2} x^{2} dx$ using 4 subintervals.


  1️⃣ Left Riemann Sum

  2️⃣ Midpoint Riemann Sum

  3️⃣ Trapezoidal Rule

  4️⃣ Right Riemann Sum
• What mathematical concept is used to represent the accumulation of change?

  Definite integral
• The integral $\int_{a}^{b} f(x) dx$ calculates the accumulated change of a function $f(x)$ from $x =$$a$ to x = b
• In the integral $\int_{a}^{b} f(x) dx$, $f(x)$ represents the rate of change.
• The definite integral $\int_{0}^{5} 3t^{2} dt$ equals 125
• What are Riemann sums used to approximate?

  Definite integrals
• Riemann sums divide the interval into subintervals and sum the areas of rectangles or trapezoids.
• What does the symbol > represent in calculus?

  Rate of change
• To find the total water accumulated from $t =$$0$ to $t =$$5$ minutes for $f(t) =$$3t^{2}$, we calculate the definite integral.
• 125 liters of water accumulated in the tank.
• What are Riemann Sums used to approximate?

  Definite integrals
• The Left Riemann Sum uses the left endpoint of each subinterval to determine the height of the rectangle.
• Match the Riemann Sum type with its characteristic:

  Left Riemann Sum ↔️ Uses left endpoint

  Right Riemann Sum ↔️ Uses right endpoint

  Midpoint Riemann Sum ↔️ Uses midpoint

  Trapezoidal Rule ↔️ Uses trapezoids
• The Trapezoidal Rule approximates the area using trapezoids instead of rectangles.
• Approximate $\int_{0}^{2} x^{2} dx$ using 4 subintervals and Riemann Sums in increasing order of approximation:


  1️⃣ Left Riemann Sum: 2.75

  2️⃣ Midpoint Riemann Sum: 3.375

  3️⃣ Trapezoidal Rule: 3.75

  4️⃣ Right Riemann Sum: 4.75
• Which Riemann Sum type gives an overestimate for $\int_{0}^{2} x^{2} dx$ using 4 subintervals?


  Right Riemann Sum
• The Fundamental Theorem of Calculus Part 1 states that if $F(x) =$$\int_{a}^{x} f(t) dt$, thenF'(x) = f(x)</latex> is the derivative of F(x).
• What does the Fundamental Theorem of Calculus Part 2 allow us to calculate?

  Definite integrals
• To find the area under $f(x) =$$x^{2}$ from $0$ to $2$, we calculate $\int_{0}^{2} x^{2} dx =$$\frac{8}{3}$, which uses the Fundamental Theorem of Calculus Part 2.
• If F(x) = \int_{0}^{x} t^{3} dt</latex>, then $F'(x) =$$x^{3}$.
• What is the formula for finding the volume of a solid using cross-sectional areas?

  $V =$$\int_{a}^{b} A(x) dx$
• What does the Fundamental Theorem of Calculus (FTC) connect?

  Differentiation and integration
• FTC Part 1 states that if F(x) = \int_{a}^{x} f(t) dt</latex>, then <latex>F'(x) = f(x)
• FTC Part 2 calculates the definite integral by finding an antiderivative $F(x)$ such that $F'(x) =$$f(x)$ and evaluating $F(b) - F(a)$.
• To find the area under $f(x) =$$x^{2}$ from $0$ to $2$, we calculate \int_{0}^{2} x^{2} dx = \frac{8}{3}
• Match the concept with its formula:

  FTC Part 1 ↔️ $F'(x) =$$f(x)$

  FTC Part 2 ↔️ $\int_{a}^{b} f(x) dx =$$F(b) - F(a)$

  Area ↔️ $A =$$\int_{a}^{b} f(x) dx$

  Volume ↔️ $V =$$\int_{a}^{b} A(x) dx$
• Steps to calculate accumulated change using a definite integral:

  1️⃣ Identify the rate of change function $f(x)$

  2️⃣ Determine the interval $[a, b]$

  3️⃣ Calculate the definite integral $\int_{a}^{b} f(x) dx$

  4️⃣ Interpret the result as the total change
• Riemann sums approximate the definite integral by dividing the interval into subintervals and summing the areas of rectangles or trapezoids.
• In a left Riemann sum, the height of each rectangle is determined by the left endpoint of the subinterval.
• Match the Riemann sum type with its formula:

  Left Riemann Sum ↔️ $\sum_{i = 1}^{n} f(x_{i - 1}) \Delta x$

  Right Riemann Sum ↔️ $\sum_{i = 1}^{n} f(x_{i}) \Delta x$
• What is the formula for the Right Riemann Sum?

  $\sum_{i = 1}^{n} f(x_{i}) \Delta x$
• The Right Riemann Sum uses the right endpoint of each subinterval to determine the height of the rectangle.
• Riemann Sums are used to approximate the value of a definite integral.
• What is the formula for the Left Riemann Sum?

  $\sum_{i = 1}^{n} f(x_{i - 1}) \Delta x$