Evaluating Using a Graphing Calculator
The calculator is especially useful in evaluating definite integrals when the x-intercepts are not
easily determined otherwise or when an explicit antiderivative of f is not obvious (or does not exist).
EXAMPLE 1
Evaluate
SOLUTION: The integrand f (x) = e−x^2 has no easy antiderivative. The calculator estimates the
value of the integral to be 0.747 to three decimal places.EXAMPLE 2
In Figure N7–5, find the area under f (x) = −x^4 + x^2 + x + 10 and above the x-axis.FIGURE N7–5
SOLUTION: To get an accurate answer for the area use the calculator to find the two
intercepts, storing them as P and Q, and then evaluate the integral:which is accurate to three decimal places.Region Bounded by a Parametric Curve
If x and y are given parametrically, say by x = f (θ), y = g(θ), then to evaluate we express y, dx,
and the limits a and b in terms of θ and dθ, then integrate. Remember that we define dx to be x ′(θ) dθ,