Cards In This Deck

• $$\begin{gathered} \text{True or False?} \\ \text{When evaluating a definite integral like }{\int }_1^3(x^2+2) \\ ,dx\text{, you must include the constant of integration }+c\text{ in the square brackets.} \end{gathered}$$
  $$\begin{gathered} \text{❌ False} \\ \text{✅ Correct: You do not need to include }+c\text{.} \end{gathered}$$
• $$\begin{gathered} \text{How do you handle a definite integral with an unknown in the limits, e.g., }{\int }_1^k(2x+1) \\ ,dx=12\text{?} \end{gathered}$$
  $$\begin{gathered} \text{1. Integrate the function algebraically.} \\ \text{2. Substitute the limits }k\text{ and }1\text{ to create an expression in }k\text{.} \\ \text{3. Set the expression equal to 12 and solve for }k\text{.} \end{gathered}$$
• $$\begin{gathered} \text{True or False?} \\ \text{If an exam question asks you to "Use calculus to evaluate }{\int }_1^4(2x+\sqrt{x}) \\ ,dx\text{", writing down the final answer from your calculator is sufficient for full marks.} \end{gathered}$$
  $$\begin{gathered} \text{❌ False} \\ \text{✅ Correct: You must show the algebraic integration.} \end{gathered}$$
• $$\begin{gathered} \text{True or False?} \\ \text{The value of the definite integral }{\int }_a^{b}y \\ ,dx\text{ is always equal to the physical area between the curve and the }x\text{-axis.} \end{gathered}$$
  $$\begin{gathered} \text{❌ False} \\ \text{✅ Correct: The integral gives the signed value.} \end{gathered}$$
• $\text{What is the correct method to find the total area between the curve }y=f(x)\text{ and the }x\text{-axis, if the curve crosses the axis between limits }x=a\text{ and }x=b\text{?}$
  $$\begin{gathered} \text{1. Find the }x\text{-coordinate of the intersection point (root), }c\text{.} \\ \text{2. Integrate separately: }{\int }_a^{c}y \\ ,dx\text{ and }{\int }_c^{b}y \\ ,dx\text{.} \\ \text{3. Add the positive magnitudes (absolute values) of both areas.} \end{gathered}$$
• $\text{What is the first step to find the finite area bounded between a curve }y=x^2\text{ and a line }y=x+2\text{ if the limits are not given?}$
  $\text{Find the points of intersection by solving the simultaneous equations.}$
• $$\begin{gathered} \text{When finding the area bounded between two curves }y=f(x)\text{ and }y=g(x)\text{, how do you determine the order of subtraction in the integral }\int (f(x)-g(x)) \\ ,dx\text{?} \end{gathered}$$
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