Final answer:
To determine how large we should choose n so that the trapezoid-rule approximation tn to the integral sin(x)dx is accurate to within 0.00001, we can use the error formula for the trapezoid rule.
Step-by-step explanation:
To determine how large we should choose n so that the trapezoid-rule approximation tn to the integral sin(x)dx is accurate to within 0.00001, we can use the error formula for the trapezoid rule:
e = -(b-a)³/12n²f''(c),
where f''(c) is the second derivative of sin(x). Since the maximum value of the second derivative of sin(x) is 1, we can use e = -(b-a)³/12n². Solving this equation for n, we get:
n = sqrt((b-a)³ / (12e)).
Substituting b = pi and a = 0, we can calculate n:
n = sqrt((pi-0)³ / (12 * 0.00001)).
Calculating this expression will give us the value of n that we need to choose.