5-8. Compute the following estimates of ∫(0 to 8) f(x) dx using the graph in the figure.

6. T(4)

Verified step by step guidance

1. Identify that T(4) refers to the Trapezoidal Rule approximation of the integral using 4 subintervals over the interval [0, 8].
2. Calculate the width of each subinterval, \( \Delta x = \frac{8 - 0}{4} = 2 \).
3. Determine the x-values at the endpoints of the subintervals: 0, 2, 4, 6, and 8.
4. From the graph, find the corresponding function values \( f(x) \) at these points: \( f(0), f(2), f(4), f(6), f(8) \).
5. Apply the Trapezoidal Rule formula: \[ T(4) = \frac{\Delta x}{2} \left[ f(0) + 2f(2) + 2f(4) + 2f(6) + f(8) \right] \] This will give the estimate of the integral \( \int_0^8 f(x) \, dx \).

Key Concepts

Definite Integral as Area Under a Curve

The definite integral of a function over an interval represents the net area between the function's graph and the x-axis. It can be approximated by summing areas of shapes like rectangles or trapezoids under the curve, which is essential for estimating integrals from graphs.

Trapezoidal Rule

The trapezoidal rule approximates the integral by dividing the interval into subintervals and approximating the area under the curve as trapezoids rather than rectangles. The formula averages the function values at the endpoints of each subinterval, providing a more accurate estimate than simple rectangular sums.

Partitioning the Interval

To apply numerical integration methods like the trapezoidal rule, the interval of integration is divided into equal or specified subintervals. For T(4), the interval [0,8] is divided into 4 subintervals of length 2, and function values at these points are used to calculate the trapezoidal sum.