6.6 Applying Properties of Definite Integrals

• What does a definite integral represent graphically?

  Area under a curve
• The multiplicativity property of definite integrals allows you to multiply the limits of integration by a constant.

  False
• The constant multiple property of definite integrals states that \int_{a}^{b} k f(x) \, dx = k \int_{a}^{b} f(x) \, dx</latex>, where $k$ is a constant
• Steps to combine and apply properties to evaluate definite integrals

  1️⃣ Identify the properties applicable to the integral

  2️⃣ Apply the linearity property if necessary

  3️⃣ Use the constant multiple property to factor out constants

  4️⃣ Apply the additivity property to split the integral

  5️⃣ Reverse limits if needed
• The linearity property of definite integrals states that $\int_{a}^{b} [f(x) +$$g(x)] \, dx =$$\int_{a}^{b} f(x) \, dx +$$\int_{a}^{b} g(x) \, dx$, which means the integral of a sum is the sum
• What is the value of $\int_{1}^{3} [3(2x) +$$x^{2}] \, dx$?


  \frac{98}{3}</latex>
• What does the additivity property of definite integrals allow you to do?

  Split the integral
• Swapping the limits of integration in a definite integral changes its value to zero.

  False
• The additivity property is useful when $f(x)$ is defined piecewise.


  True
• The additivity property allows breaking up a definite integral into smaller intervals
• What is the value of $\int_{0}^{2} x^{2} \, dx$?


  $\frac{8}{3}$
• What is the formal definition of a definite integral using Riemann sums?

  \int_{a}^{b} f(x) \, dx = \lim_{n \to \infty} \sum_{i = 1}^{n} f(x_{i}^ * ) \Delta x</latex>
• Reversing the limits of integration changes the sign
• The additivity property can simplify complex integrals by splitting them into manageable parts.


  True
• What is the formula for the additivity property of definite integrals?

  $\int_{a}^{c} f(x) \, dx =$$\int_{a}^{b} f(x) \, dx +$$\int_{b}^{c} f(x) \, dx$
• What is the formula for the additivity property of definite integrals?

  $\int_{a}^{c} f(x) \, dx =$$\int_{a}^{b} f(x) \, dx +$$\int_{b}^{c} f(x) \, dx$
• The additivity property states that the total area under a curve equals the sum of areas from $a$ to $b$ and $b$ to $c$.


  True
• The additivity property is also referred to as the interval
• What is the correct name for the property that allows factoring out constants from integrals?

  Constant multiple rule
• The constant multiple rule can be used to simplify integral calculations by handling constant factors separately.


  True
• Swapping the limits of integration changes the sign of the integral.
• The reversing limits property allows calculations when the lower limit is greater than the upper limit.


  True
• A definite integral represents the area under a curve between two specified points
• Constants can be factored out of a definite integral using the constant multiple property
• Steps to calculate the definite integral $\int_{1}^{3} (2x + x^{2}) \, dx$


  1️⃣ Apply linearity to split the integral

  2️⃣ Find the antiderivatives of 2x and $x^{2}$

  3️⃣ Evaluate the antiderivatives at the limits

  4️⃣ Calculate the final result
• Why is the additivity property useful when calculating definite integrals?

  Simplifies complex calculations
• What is the formula for the additivity property of definite integrals?

  \int_{a}^{c} f(x) \, dx = \int_{a}^{b} f(x) \, dx + \int_{b}^{c} f(x) \, dx</latex>
• The additivity property of definite integrals states that \int_{a}^{c} f(x) \, dx = \int_{a}^{b} f(x) \, dx + \int_{b}^{c} f(x) \, dx</latex>, where $b$ lies within (a, c)
• What happens to a definite integral when its limits are reversed?

  Changes sign
• The order property of definite integrals states that if $f(x) > 0$ for all $x$ in $[a, b]$, then $\int_{a}^{b} f(x) \, dx > 0$.


  True
• How is a definite integral formally defined using Riemann sums?

  \int_{a}^{b} f(x) \, dx = \lim_{n \to \infty} \sum_{i = 1}^{n} f(x_{i}^ * ) \Delta x</latex>
• Swapping the limits of integration in a definite integral changes its sign.


  True
• The linearity property of definite integrals applies only to sums of two functions.

  False
• The constant multiple property of definite integrals states that $\int_{a}^{b} k f(x) \, dx =$$k \int_{a}^{b} f(x) \, dx$, allowing you to factor out the constant
• What is the formula for the additivity property of definite integrals?

  $\int_{a}^{c} f(x) \, dx =$$\int_{a}^{b} f(x) \, dx +$$\int_{b}^{c} f(x) \, dx$
• What is the value of $\int_{0}^{4} x^{2} \, dx$ using $b =$$2$?


  $\frac{64}{3}$
• What does the additivity property of definite integrals state?

  $\int_{a}^{c} f(x) \, dx =$$\int_{a}^{b} f(x) \, dx +$$\int_{b}^{c} f(x) \, dx$
• To calculate $\int_{0}^{4} x^{2} \, dx$ using $b =$$2$, we can split the integral into $\int_{0}^{2} x^{2} \, dx$ and $\int_{2}^{4} x^{2} \, dx$


  True
• A definite integral represents the area under a curve
• Match the property of definite integrals with its explanation:

  Linearity ↔️ The integral of a sum is the sum of the integrals

  Additivity ↔️ The integral over an interval can be split into subintervals

  Constant Multiple ↔️ Constants can be factored out of integrals

  Reversing Limits ↔️ Swapping the limits changes the sign