1.8 Integration

• The fundamental theorem of calculus states that integration "undoes" differentiation.


  True
• The symbol ∫ in integration represents the integral.


  True
• Steps to apply the power rule for integration to ∫x^n dx

  1️⃣ Add 1 to the exponent

  2️⃣ Divide by the new exponent

  3️⃣ Add the constant of integration
• The constant of integration in integration is denoted by C
• The power rule for integration is used to find the antiderivative of a function in the form x^n
• The constant of integration is denoted by the letter C.

  True
• Steps to integrate the polynomial function \(\int(2x^3 - 4x^2 + 3x - 1) \, dx\)

  1️⃣ Integrate \(2x^3\): \(\frac{2x^4}{4} + C\)

  2️⃣ Integrate \(-4x^2\): \(-\frac{4x^3}{3} + C\)

  3️⃣ Integrate \(3x\): \(\frac{3x^2}{2} + C\)

  4️⃣ Integrate \(-1\): \(-x + C\)

  5️⃣ Combine the results: \(\int(2x^3 - 4x^2 + 3x - 1) \, dx = \frac{2x^4}{4} - \frac{4x^3}{3} + \frac{3x^2}{2} - x + C\)
• The constant of integration is added to the antiderivative because there are infinitely many functions with the same derivative.


  True
• The constant of integration (C) is added to the antiderivative because there are infinitely many functions that could have the same derivative
• Match the component of integration with its description:

  Integrand ↔️ The function being integrated

  Variable of integration ↔️ Specifies the integration variable

  Limits of integration ↔️ Interval over which integration occurs

  Constant of integration ↔️ Accounts for infinite antiderivatives
• The power rule for integration states that $\int x^{n} \, dx =$$\frac{x^{n + 1}}{n + 1} +$$C$


  True
• To integrate polynomials, the power rule is applied to each term separately.

  True
• The constant of integration is always included in indefinite integrals.


  True
• The integral of $5x^{2}$ is $\frac{5}{3}x^{3} +$$C$.


  True
• What does a definite integral calculate?

  Area under a curve
• What does a definite integral calculate?

  Area under a curve
• What theorem is used to evaluate definite integrals?

  Fundamental theorem of calculus
• How do you find the value of a definite integral after finding its antiderivative?

  Subtract \( F(a) \) from \( F(b) \).
• Integration is the reverse process of differentiation.


  True
• What is another name for the antiderivative of a function?

  Function
• What is the goal of integration in calculus?

  Recover original function
• What is integration the reverse process of?

  Differentiation
• The power rule for integration involves finding the antiderivative of a function in the form x^n
• What is the constant of integration in the power rule formula?

  C
• The antiderivative of ∫x^3 dx is x^4/4 + C.

  True
• What is another name for the original function recovered through integration?

  Antiderivative
• How do you integrate a polynomial function using the power rule?

  Apply the rule to each term
• What is the general formula for integrating \(ax^n\)?

  $\int ax^{n} \, dx =$$\frac{a}{n + 1} x^{n + 1} +$$C$
• What are the key components of integral notation (∫)?

  Integrand, variable, limits, constant
• The indefinite integral of $x^{2}$ with respect to $x$ is \frac{x^{3}}{3} + C</latex>


  True
• Arrange the steps to demonstrate the concept of integration as the reverse of differentiation:

  1️⃣ Differentiate a function, e.g., $f(x) =$$x^{2} +$$3$

  2️⃣ Obtain the derivative, e.g., $f'(x) =$$2x$

  3️⃣ Integrate the derivative, e.g., $\int 2x \, dx$

  4️⃣ Recover the original function with a constant, e.g., $x^{2} +$$C$
• Integrating $2x$ results in $x^{2} +$$C$, where $C$ is the constant of integration
• What is the antiderivative of x^{3}</latex> using the power rule for integration?

  $\frac{x^{4}}{4} +$$C$
• What is the integral of the polynomial $3x^{2} - 5x +$$2$?


  $x^{3} - \frac{5}{2}x^{2} +$$2x +$$C$
• What is the general formula for integrating a function with a constant coefficient?

  $\int ax^{n} \, dx =$$\frac{a}{n + 1} x^{n + 1} +$$C$
• What is the symbol for integration?

  $\int \, dx$
• Definite integrals include the constant of integration.

  False
• A definite integral yields a numerical value as its outcome.


  True
• The fundamental theorem of calculus relates differentiation and integration.


  True
• Integration can be used to calculate the volume of a solid of revolution.


  True