Welcome back, students! We are approaching the Chapter 5 Test, which means it is time to solidify your understanding of Integration. This chapter marks a major shift from differentiation to integration, and mastering these concepts is crucial for the rest of your calculus journey.

I have uploaded two important documents to help you study: the Review Sheet (blank questions) and the Answer Sheet (handwritten solutions). I highly recommend trying the problems on your own first before checking the answers.

Key Concepts Covered in Chapter 5

Based on the review sheet, here are the specific topics you need to be comfortable with:

  • Riemann Sums & Area Approximation: You will need to estimate the area under a curve using rectangles. Remember to check if the problem asks for a lower estimate or an upper estimate based on the graph's behavior.
    Example: Using 5 rectangles to estimate the area from $x=0$ to $x=10$.
  • Definite Integrals as Area: Understand the geometric interpretation of the integral $\int_a^b f(x) dx$. Remember that area below the x-axis is considered negative in a net integral calculation.
  • The Fundamental Theorem of Calculus:
    • Part 1: Finding the derivative of an accumulation function. For example, if $F(x) = \int_0^x \frac{t^2}{1+t^3} dt$, then $F'(x) = \frac{x^2}{1+x^3}$.
    • Part 2: Evaluating definite integrals using antiderivatives: $\int_a^b f(x) dx = F(b) - F(a)$.
  • Initial Value Problems: You will be given information like $f''(x) = -2 + 12x - 12x^2$ along with initial conditions ($f(0)=4$, etc.) and asked to work backward to find the original function $f(x)$. Don't forget your constant of integration, $+C$!
  • Techniques of Integration:
    • Power Rule: $\int x^n dx = \frac{x^{n+1}}{n+1} + C$
    • U-Substitution: This is vital for integrals like $\int_1^5 \frac{dt}{(t-4)^2}$ or $\int \frac{x+2}{\sqrt{x^2+4x}} dx$. Identify your $u$, find $du$, and substitute carefully.

Application: Particle Motion

One of the most important applications in this chapter is describing particle motion. You will be given a velocity function, such as $v(t) = t^2 - t$. You must distinguish between:

  1. Displacement: The net change in position, calculated as $\int_0^5 v(t) dt$.
  2. Total Distance Traveled: The total ground covered, calculated as $\int_0^5 |v(t)| dt$.

Tip: To find total distance, you must find where the particle stops (where $v(t)=0$), break the integral into pieces, and ensure all area contributions are positive.

Study Strategy

Download the Chapter 5 Test Review Sheet attached below and simulate a test environment. Give yourself time to work through the integrals without looking at your notes. Once you are finished, compare your work with the Answer Sheet. Pay close attention to how the $u$-substitution boundaries are handled in definite integrals!

Good luck studying! You have all the tools you need to succeed.