To compute integrals involving functions like $f(t)$ or $g(t)$ as given in the problem, here’s a step-by-step overview:

(1) Understand the Graph and Function Behavior

• Identify how the functions $f(t)$ and $g(t)$ behave over the specified intervals.
• Break the interval of integration into subintervals where the function is either linear or constant. For example, you might divide the interval into segments like $[2, 4], [4, 6]$, etc., based on changes in slope or behavior.

(2) Evaluate the Definite Integral

• For each subinterval, compute the area between the graph and the $t$-axis.
• The integral is the net area:
  • Areas above the $t$-axis contribute positively.
  • Areas below the $t$-axis contribute negatively.
• If necessary, account for shifts (e.g., subtracting 6 in part (a)).

(3) Decompose Composite Expressions

• When integrating expressions like $f(t) - g(t)$, break it into: $\int f(t) \, dt - \int g(t) \, dt$
• Compute each part separately and combine the results.

(4) Apply Fundamental Theorem of Calculus

• Use this theorem when dealing with a defined function like $H(x) = \int_{8}^{x} f(t) \, dt$: $H'(x) = f(x)$
• To find $H(10)$, evaluate: $H(10) = \int_{8}^{10} f(t) \, dt$
• This means calculating the area under $f(t)$ from $t = 8$ to $t = 10$.

(5) Set Up and Solve for Each Part

• (a) Calculate $\int_{2}^{0} (f(t) - 6) \, dt$:
  • Adjust for the negative order of limits by reversing the integral: $\int_{2}^{0} = -\int_{0}^{2}$
  • Subtract 6 from $f(t)$ and compute the integral based on the graph.
• (b) Calculate $\int_{2}^{6} (f(t) - g(t)) \, dt$:
  • Find the area difference between $f(t)$ and $g(t)$ over $[2, 6]$.
• (c) Compute $H(10) = \int_{8}^{10} f(t) \, dt$:
  • Focus on the graph of $f(t)$ over the interval $[8, 10]$.

General Tips for Solving These Problems:

1. Graph Interpretation: Pay close attention to slopes and critical points on the graph.
2. Break Down Intervals: Compute areas in manageable sections.
3. Consider the Signs: Watch for when the curve dips below the $t$-axis (negative contributions to area).
4. Use Symmetry: If graphs have symmetry, simplify computations by leveraging it.

Would you like me to work through each part of this specific problem with step-by-step calculations?