What is the estimated area under the curve of f(t) = t^2 from t = 0 to t = 2 using the trapezoidal rule with delta t = 0.5?

To find the estimated area under the curve of the function f(t) = t^2 from t = 0 to t = 2 using the trapezoidal rule with a delta t of 0.5, we first identify the points at which we will evaluate the function. With delta t set to 0.5, the intervals from t = 0 to t = 2 are:

• t = 0

• t = 0.5

• t = 1.0

• t = 1.5

• t = 2.0

Next, we calculate the value of the function at each of these points:

• f(0) = 0^2 = 0

• f(0.5) = (0.5)^2 = 0.25

• f(1.0) = (1.0)^2 = 1

• f(1.5) = (1.5)^2 = 2.25

• f(2.0) = (2.0)^2 = 4

Now, applying the trapezoidal rule, we estimate the area as follows. According to the trapezoidal rule, the area