Calculus- Intergration

• What skill is covered in section 1.11?

  Sketching the derived function
• What two conditions are to be determined when sketching a derived function?

  Where the gradient, m, is +ve and -ve
• How does the gradient relate to the graph of the derivative function?

  • Gradient is +ve when $f'(x)$ is above the x-axis
  • Gradient is -ve when $f'(x)$ is below the x-axis
• What is the main topic covered in section 12?

  Calculus 2 - Integration
• What is the title of section 12.1?

  Basic Integration
• What power rule is followed when integrating terms?

  • Add one to the power
  • Divide by the new power
• What is the result of integrating $x^n dx?$


  $\frac{x^{n+1}}{n+1} +$$C$
• When integrating indefinitely, what value is added to the integral?

  The constant of integration, $C$
• What are the steps involved in preparing a function for integration?

  1. Break the function down into individual terms
  2. Express each term on the numerator of any fraction
  3. Express each term in index form (see section 11.1)
• What is the title of section 12.3?

  Definite integrals
• What are the two steps involved in evaluating a definite integral?

  1. Integrate the function
  2. Evaluate between two limits
• What is the result of evaluating $\int_a^b x^2 dx?$


  $\left[\frac{x^3}{3}\right]_a^b =$$\frac{b^3}{3} - \frac{a^3}{3}$
• What is the title of section 12.4?

  Area under curves
• What are the steps in finding the total area under a curve?

  1. Find the area above the x-axis
  2. Find the area below the x-axis (ignore the negative)
  3. Add the areas together
• What page number is Zeta Maths Limited referred to?

  15
• What notation is used to represent a definite integral between limits a and b?

  $\int_a^b f(x) dx$
• How should areas below the x-axis be written when evaluating definite integrals?

  As positive values
• What are the steps in evaluating a definite integral with curves?

  1. Set the curves equal to each other and solve to find the limits
  2. Set up the integral as $\int_a^b [f(x) - g(x)] dx$ where $f(x)$ is the upper curve and $g(x)$ is the lower curve
  3. Evaluate the integral
• What are equations of the form $\frac{dy}{dx} =$$ax +$$b$ called?


  Differential equations
• How are differential equations solved?

  By integration
• If $y =$$3x^3 +$$C$ passes through (1,5), what is the value of $C?$


  $C =$$2$
• What topic is covered in section 12.6?

  Differential equations
• What is the title of section 12.6?

  Differential equations